Conjugacy growth of commutators
Peter S. Park


Conjugacy Growth of Commutators]Conjugacy Growth of Commutators



Peter S. Park]Peter S. Park
Department of Mathematics, Harvard University, Cambridge, MA 02138
mailto:pspark@math.harvard.edupspark@math.harvard.edu




For the free group F_r on r>1 generators (respectively, the free product  G_1 * G_2 of two nontrivial finite groups G_1 and G_2), we obtain the asymptotic for the number of conjugacy classes of commutators in F_r (respectively, G_1 * G_2) with a given word length in a fixed set of free generators (respectively, the set of generators given by the nontrivial elements of G_1 and G_2). Our result is proven by using the classification of commutators in free groups and in free products by Wicks, and builds on the works of Rivin and Sharp, who asymptotically counted the conjugacy classes of commutator-subgroup elements in F_r with a given word length. 

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    December 30, 2023
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§ INTRODUCTION

Let G be a finitely generated group with a finite symmetric set of generators . Any element g∈ G can then be written as a word in the letters of , and one can  define the length of g by

    inf{  k : ∃ c_1,…,c_k s.t. g=c_1… c_k}.

Consider the closed ball B_k(G,) ⊂ G of radius k in the word metric, defined as the subset consisting of elements with length ≤ k. One can then ask natural questions about the growth of G: how large is |B_k(G,)| as k →∞, and more generally, what connections can be made between the properties of G and this notion of its growth rate? For example, one of the pioneering results on this question is that of Gromov <cit.>, who classified the groups G with polynomial growth.

Since the middle of the 20th century, the growth of groups has been widely studied in various contexts largely arising from geometric motivations, such as characterizing the volume growth of Riemannian manifolds and Lie groups. In fact, in addition to applying knowledge on the growth of groups to describe growth in such geometric settings, one can also pass information in the other direction, i.e., use information on volume growth in geometric settings to yield consequences regarding the growth of the relevant groups. For instance, Milnor <cit.> used inequalities relating the volume and curvature of Riemannian manifolds to prove that the fundamental group of any compact, negatively-curved Riemannian manifold has at least exponential growth. 

A group that arises especially commonly in such geometric contexts is the free group F_r on r>1 generators, which also has exponential growth; more precisely, after fixing a symmetric generating set {x_1,…,x_r,x_1^-1, …, x_r^-1}, it is easy to see that 

    |B_k(G,)| = 1+∑_i=1^k |∂ B_i(G,)|    = 1+ ∑_i=1^k 2r(2r-1)^i-1
       = 1+r((2r-1)^k-1)/r-1,

where ∂ B_k(G,) denotes the subset of length-k elements.



 §.§ Motivation and main results

In certain contexts, it is more natural to consider the growth rate of the conjugacy classes of G. For a given conjugacy class  of G, define the length of  by

    inf_g ∈(g),

and define ∂ B_k^(G,) as the set of conjugacy classes of G with length k.  In the case of F_r, the minimal-length elements of a conjugacy class are precisely its cyclically reduced elements, all of which are cyclic conjugates of each other. The conjugacy growth of F_r can be described as ∂ B_k^(G,) ∼ (2r-1)^k/k, which agrees with the intuition of identifying the cyclic conjugates among the 2r(2r-1)^k-1 words of length k; for the full explicit formula, see <cit.>. 

One context for which conjugacy growth may be a more natural quantity to study than the growth rate in terms of elements is when characterizing the frequency with which a conjugacy-invariant property occurs in G. An example of such a property is membership in the commutator subgroup [G,G]. On this front, Rivin <cit.> computed the number c_k of length-k cyclically reduced words in F_r that are in the commutator subgroup (i.e., have trivial abelianization) to be the constant term in the expression

    (2√(2r-1))^k T_k( 1/2√(2r-1)∑_i=1^r (x_i+ 1/x_i)),

where T_k denotes the kth Chebyshev polynomial of the first kind. Furthermore, Sharp <cit.> obtained the asymptotics of c_k, given by

    c_2m∼4r(2r-1)^2m-1/(2π)^r/2σ^r m^r/2,

where 

    σ(1/√(2r-1)( 1+(r+√(2r-1)/r-√(2r-1))^1/2))^1/2;

note also that when k is odd, we have c_k=0. From the number of cyclically reduced words with trivial abelianization, one can derive the  growth of  conjugacy classes with trivial abelianization by using Möbius inversion, due to the following relationships:

    c_k = ∑_d | k p_d,

where p_d denotes the number of primitive (i.e., not a proper power of any subword) length-d words with trivial abelianization, and 

    |∂ B_k^(G,) ∩ [G,G]| = ∑_d | kp_d/d,

which together imply by Möbius inversion that

    |∂ B_k^(G,) ∩ [G,G]| = ∑_d | k1/d∑_e | dμ( d/e) c_e    = ∑_e | kc_e/e( ∑_d' |k/eμ(d')/d')
        = ∑_e | kc_e/e·ϕ(k/e)/k/e =  ∑_e | k c_eϕ(k/e)/k.

In the above, ϕ denotes the Euler totient function. For details on this derivation, the reader is directed to <cit.> .


In this paper, we answer the analogous question for commutators rather than for commutator-subgroup elements. This new inquiry is structurally different in that it aims to solve a Diophantine equation over a group G (whether, for a given W ∈ G, there exist X,Y∈ G such that XYX^-1Y^-1=W), rather than a subgroup-membership problem (whether W is in [G,G]). In particular, the set of commutators is not multiplicatively closed, so we cannot use primitive words as a bridge between counting cyclically reduced words and counting conjugacy classes as above. Instead, we use a theorem of Wicks <cit.>, which states that an element of F_r is a commutator if and only if it is a cyclically reduced conjugate of a commutator satisfying the following definition.

A Wicks commutator of F_r is a cyclically reduced word  W∈ F_r of the form ABCA^-1B^-1C^-1. Equivalently, it is a word of the form ABCA^-1B^-1C^-1 such that the subwords A,B, and C are reduced; there are no cancellations between the subwords A,B,C,A^-1,B^-1, and C^-1; and the first and last letters are not inverses.

Using this classification, we count the number of conjugacy classes of commutators in F_r with length k  by counting the number of Wicks commutators with length k.


Let k ≥ 0 be even. The number of distinct conjugacy classes of commutators  in F_r with length k is given by

    (2r-2)^2(2r-1)^k/2-1/96r(k^2 + O_r(k)  ),

where the implied constant  depends only on r and is effectively computable.

Note that the number  of conjugacy classes of commutators in F_r is roughly proportional to the square root of the number (<ref>) of all conjugacy classes with trivial abelianization.

We also employ a similar argument, using Wicks' characterization of commutators in free products, to answer the analogous question for the free product G_1 * G_2 of two nontrivial finite groups. Like free groups, free products have exponential growth, which has been studied recently by Mann <cit.>. This group is of independent interest as the isomorphism class of _2(); specifically, for the usual generators

    S[  0 -1;  1  0 ] and  T [ 1 1; 0 1 ]

of _2(), we have that S corresponds to a generator of the G_2 factor and ST, to a generator of the G_1 factor. We consider the set of generators  (G_1 ∖{1}) ∪  ( G_2 ∖{1} ) of G_1 * G_2. Then, a theorem of Wicks <cit.> analogous to the previous one implies that an element of G_1 * G_2 is  a commutator if and only if it is a cyclic conjugate of a commutator satisfying the following definition.


A Wicks commutator of G_1 *G_2 is a word W∈ G_1 * G_2 that is fully cyclically reduced, which is to say that the adjacent letters (i.e., nonidentity elements of G_1 and G_2 in the word) are in different factors of the free product, as are the first and last letters; and in one of the following forms:


  * ∈ G_1 that is a commutator of G_1,

  * ∈ G_2 that is a commutator of G_2,

  * A _1 A _2^-1 for _1, _2 ∈ G_2 that are conjugates,

  * _1  A_2^-1 A for _1, _2 ∈ G_1 that are conjugates,

  * ABA^-1B^-1,

  * A_1 B _2  A^-1_3  B^-1_4 for _1, _2, _3, _4 ∈ G_2 satisfying _4_3_2_1=1,

  * _1 A _2  B _3  A^-1_4  B^-1 for _1, _2, _3, _4 ∈ G_1 satisfying _4_3_2_1=1,

  * A_1 B _1 C_2  A^-1_2  B^-1_3 C^-1_3 for _1, _2, _3, _1, _2, _3∈ G_2 satisfying _3_2_1=1 and _3_2_1=1, with A, B, and C nontrivial,

  * _1 A_1 B _2 C_2  A^-1_3  B^-1_3 C^-1 for _1, _2, _3 ,_1, _2, _3∈ G_1 satisfying _3_2_1=1 and _3_2_1=1, with A, B, and C nontrivial.



A fully cyclically reduced element W∈ G_1 *G_2 with length k>1  alternates between k/2 letters in  and k/2 letters in , where k/2 is necessarily an integer. Thus, if k>1 is odd, then there are no fully cyclically reduced elements of G_1 * G_2 with length k. Furthermore, for W to be a Wicks commutator not of the form (1) or (2) in  Definition <ref>,  it is necessary not only that k is even, but also that k/2 is an even integer. Thus, all  but possibly finitely many (the length-1 exceptions) Wicks commutators of G_1*G_2 have length divisible by 4. For any k divisible by 4, we obtain the number of length-k conjugacy classes in G_1 * G_2 comprised of commutators.

Let k ≥ 0 be a multiple of 4. The number of distinct conjugacy classes of commutators in G_1 * G_2 with length k is given by

    1/192(( |G_2|-2 )^2 +( |G_1|-2 )^2 )     k^2  ^k/4-1^k/4-1
       +O_|G_1|,|G_2|( k^k/4^k/4),

where the implied constant only depends on |G_1| and |G_2|, and is effectively computable.


Suppose that 4 | k> 0. Then, the set of cyclically reduced elements of G_1 * G_2 (whose letters of odd position are elements of G_1, without loss of generality) with length k and trivial abelianization bijectively maps to the Cartesian product of two sets: the set of closed paths of length k/2 on the complete graph K_|G_1| with fixed basepoint P_1, and the set of closed paths of length k/2 on the complete graph K_|G_2| with fixed basepoint P_2. For a graph E, the number of closed paths on E with length n is given by  ∑λ^n, where λ ranges over the eigenvalues (counted with multiplicity) of the adjacency matrix of E. For m≥ 2, the adjacency matrix of K_m is the m× m matrix with diagonal entries 0 and all other entries 1; its eigenvalues are m-1 (with multiplicity 1) and -1 (with multiplicity m-1). Consequently, for even integers n>0, the number of closed paths on K_m with length n is given by (m-1)^n+(m-1)(-1)^n=m(m-1)^n-1. By symmetry between the m vertices, the number of such closed paths with a given basepoint is (m-1)^n-1. 
    (m-1)^n+(m-1)(-1)^n/m,

which is equal to 1 for m=2 and can be written as 

    (m-1)^n/m +O(1)

for m>2.Thus, the number of cyclically reduced words with length k and trivial abelianization in  G_1 * G_2 is given by

    (|G_1|-1)^k/2-1(|G_2|-1)^k/2-1.


    {[ (|G_1|-1)^n/|G_1| +O(1),              if  |G_1|>2;                       1,              if  |G_1|=2 ]}·{[ (|G_2|-1)^n/|G_2| +O(1),              if  |G_2|>2;                       1,              if  |G_2|=2 ]},
Applying Möbius inversion as done in (<ref>), we see that the number of conjugacy classes of commutators in G_1 * G_2 with length k is roughly comparable to the square root of the number of all length-k conjugacy classes with trivial abelianization.



 §.§ Geometric applications

Counting conjugacy classes of commutators has a topological application. Let X be a connected CW-complex with fundamental group π_1(X), and let  be a conjugacy class of π_1(X) with trivial abelianization, corresponding to the free homotopy class of a homologically trivial loop γ : S^1  → X. Then, the commutator length of , defined as the minimum number of commutators whose product is equal to an element of , is also the minimum genus of an orientable surface (with one boundary component) that continuously maps to X so that the boundary of the surface maps to γ <cit.>. Thus, a conjugacy class of π_1(X) is comprised of commutators if and only if its corresponding free homotopy class γ: S^1 → X
satisfies the following topological property:

    There exists a torus Y  with one boundary component ⋆







and a continuous map f : Y → X satisfying f(∂ Y)=γ.





Consequently, we obtain the following. 

Let X be a connected CW-complex.


  * Suppose X has fundamental group F_r with a symmetric set of free generators . Then, the number of free homotopy classes of   loops γ : S^1 → X with length k (in the generators of ) satisfying Property (⋆) is given by

    (2r-2)^2(2r-1)^k/2-1/96r(k^2 + O_r(k )  ),
 
where the implied constant  depends only on r and is effectively computable.


  * Suppose X has fundamental group G_1 * G_2 with the set of generators 
    (G_1 ∖{1}) ∪ (G_2 ∖{1}).
 
 Then, the number of free homotopy classes of   loops γ : S^1 → X with length k (in the generators of ) satisfying Property (⋆)  is given by

    1/192(( |G_2|-2 )^2 +( |G_1|-2 )^2 )
        k^2  ^k/4-1^k/4-1
      + O_|G_1|,|G_2|   ( k^k/4^k/4),

where the implied constant depends only on |G_1| and |G_2|, and is effectively computable. 


 
For a connected CW-complex with a natural geometric definition of length for loops, one expects the word length of a free homotopy class of loops to correlate with the geometric length of the free homotopy class, taken to be the infimum of the geometric lengths of its loops, in some way. A trivial example of this phenomenon is in the case of a wedge ⋁_j=1^r S^1 of r unit circles, for which there is perfect correlation; a free homotopy class in this space has word length k if and only if the minimal geometric length of a loop in the class is k. 

A more interesting example of the connection between word length and geometric length arises from hyperbolic geometry. Let X≅Γ\ be a hyperbolic orbifold, where Γ⊂_2() denotes a Fuchsian group. Then, every free homotopy class of loops in X that does not wrap around a cusp has a unique geodesic representative, so the geometric length of the class is realized as the geometric length of this unique closed geodesic <cit.>. In particular, we have a canonical correspondence between hyperbolic free homotopy classes of loops (i.e., ones that do not wrap around a cusp) and closed geodesics. Let us specialize to the case that X is a pair of pants (surface obtained by removing three disjoint open disks from a sphere). We have that X≅Γ\ for Γ≅ F_2, so fix a symmetric set of free generators. Then, Chas, Li, and Maskit <cit.> have conjectured from computational evidence that the geometric length and the word length of a free homotopy class of loops are correlated in the following fundamental way.

For an arbitrary hyperbolic metric ρ on the pair of pants that makes its boundary geodesic, let 
_k denote the set of free homotopy classes with word length k in a fixed choice of free generators for the fundamental group. Then, there exist constants μ and σ (depending on ρ) such that for any a<b, the proportion of   ∈_k such that the geometric length h() of the unique geodesic representative of  satisfies

    h()-μ k/√(k)∈ [a,b]

converges to 

    1/σ√(2π)∫_a^b e^ -x^2/2σ^2dx

as k→∞. In other words, as k→∞, the distribution of geometric lengths g() for ∈_k approaches the Gaussian distribution with  mean μ k and standard deviation σ k.

It is natural to expect that even when  restricting to  commutators, the resulting distribution would still exhibit a fundamental correlation between geometric length and word length. If this were proven true, then one could use Corollary <ref> to indirectly count closed geodesics in X with geometric length ≤ L satisfying Property (⋆).

For a general hyperbolic orbifold Γ\, we have the aforementioned natural correspondence between the closed geodesics of Γ\ and the hyperbolic conjugacy classes of Γ. In the case that Γ is finitely generated, we claim that the number of non-hyperbolic conjugacy classes of commutators with word length k is small. Indeed, Γ only has finitely many elliptic conjugacy classes. Moreover, Γ only has finitely many primitive parabolic conjugacy classes, corresponding to equivalence classes of cusps. Denoting the primitive parabolic conjugacy classes by {_i}_1≤ i ≤ n, we can conjugate  each _i by elements in Γ to assume without loss of generality that the representative g_i is fully cyclically reduced. Then, since every parabolic conjugacy class is a power of a primitive one, we see that the number of parabolic classes with word length k is at most n. 

Suppose further that Γ is isomorphic to F_r (respectively, to G_1 * G_2 for two nontrivial finite groups G_1 and G_2) and fix a set of generators. We see that the number of non-hyperbolic conjugacy classes of commutators with length k is bounded by a constant, and thus clearly dominated by the error term of ≪ k(2r-1)^k/2 (respectively, ≪ k(|G_1|-1)^k/4(|G_2|-1)^k/4).  In fact, it is known <cit.> that a commutator of a free group cannot be a proper power, which shows that the number of non-hyperbolic conjugacy classes of commutators is finite in this case. For the latter case of Γ≅ G_1 * G_2, the work of Comerford, Edmunds, and Rosenberger <cit.> gives the possible forms that a proper-power commutator can take in a general free product. After discarding the O(1) number of non-hyperbolic conjugacy classes of commutators with word length k, one obtains from Corollary <ref> the asymptotic number of closed geodesics γ with word length k satisfying Property (⋆), with the error term unchanged. It is natural to expect for Γ\ a correlation between the word length and the geometric length similar to that conjectured for the pair of pants. If such a correlation were proven true, then Corollary <ref> would yield information on the number of  closed geodesics in Γ\ with length ≤ L satisfying Property (⋆).

For an example of such a hyperbolic orbifold Γ\ whose fundamental group Γ is isomorphic to G_1 *G_2, we introduce the Hecke group H() for a real number >0, defined by the subgroup of _2() generated by 

    S [  0 -1;  1  0 ] and  T_[ 1  ; 0 1 ].

Denote _q  2cos(π/q) for integers q ≥ 3. Hecke <cit.> proved that H() is Fuchsian if and only if =_q for some integer q ≥ 3 or ≥ 2. Specializing to the former case, H(_q) has the presentation

    ⟨ S, R_ : S^2 = R_^q = I ⟩≅/2 * /q,

where the generator R_ can be taken to be  S T_ <cit.>. Thus, our desired example H(_q) \ has fundamental group isomorphic to /2 * /q. In particular, note that H(_3) = _2() (with the standard generators S and T_1), which has a presentation as the  free product /2 * /3. 


The Chas–Li–Maskit conjecture has recently been proven in a preprint <cit.> by  Gekhtman,  Taylor, and Tiozzo.





 §.§ Connections to number theory

Counting closed geodesics on hyperbolic orbifolds Γ\ by geometric length  is a well-studied problem that often has analogies (in both the asymptotic of the resulting counting function and the techniques used to prove it) to counting problems in number theory. To illustrate, the primitive hyperbolic conjugacy classes of Γ correspond precisely to the primitive closed geodesics of Γ\. These are called prime geodesics because when ordered by trace, they satisfy equidistribution theorems analogous to those of prime numbers, such as the prime number theorem (generally credited to Selberg <cit.>, while its analogue for surfaces of varying negative curvature was proven by Margulis <cit.>) and Chebotarev's density theorem (proven by Sarnak <cit.>).  Specifically, this analogue of the prime number theorem is called the prime geodesic theorem, which states that the number of prime geodesics γ∈Γ\ with norm ≤ N is asymptotically given by ∼ N/log N, where the norm n(γ) denotes the real number ρ > 1 such that the Jordan normal form of the _2()-matrix corresponding to ℓ is [    ±ρ     0;     0 ±ρ^-1 ], which is related to the geometric length ℓ(γ) by the relation (see <cit.>) given by

    ℓ(γ)=log n(γ).

( coshℓ(γ)/2 + √(( coshℓ(γ)/2)^2-1) )^2.
Furthermore, Sarnak's analogue of Chebotarev's density theorem implies that for every surjective homomorphism φ: Γ→ G onto a finite abelian group  G, the number of prime geodesics of Γ\ with norm ≤ N in the inverse image of a given g ∈ G is asymptotically given by ∼ N/(|G| log N). In particular, if [Γ, Γ] is of finite index in Γ, then letting ϕ be the quotient map Γ→Γ/[Γ.Γ], one sees that the number of prime geodesics of Γ\ with norm ≤ N and trivial abelianization is asymptotically given by 

    ∼N/|Γ : [Γ ,Γ]| log N.

The result of Corollary <ref> can be seen as the answer to an analogous problem of counting closed geodesics of Γ\ that correspond to conjugacy classes of commutators rather than arbitrary conjugacy classes with trivial abelianization, the main difference being that our formula counts closed geodesics ordered by word length.

We mention one more number-theoretic setting which involves counting conjugacy classes of commutators of a Fuchsian group, specialized to the modular group Γ = _2() = H(_3). A commutator of Γ is precisely a coset {C, -C} for a commutator C=ABA^-1B^-1 of _2(), where A, B ∈_2(); this choice is unique because -I has nontrivial abelianization, which implies that -C does, as well. Thus, a conjugacy class of commutators in Γ  with such a representative {C, -C}  is the union of the _2()-conjugacy class of the _2()-commutator C and the _2()-conjugacy class of -C.  

For the free group F_2 on generators X and Y, the map (F_2, _2()) →^3 defined by ρ↦ (ρ(X), ρ(Y),ρ(XY)) induces an isomorphism between the moduli space of _2-valued representations of F_2 and the affine space ^3, by the work of Vogt <cit.> and Fricke <cit.>. Labeling this parametrization of  ^3 as (x,y,z), we note that  ρ(XYX^-1Y^-1) is given by the polynomial M(x,y,z) x^2+y^2+z^2-xyz-2; in other words, we have the trace identity <cit.>

    (A)^2+ (B)^2+  (AB)^2 - ( A)  (B) (AB)-2 =(ABA^-1B^-1)

for all A,B ∈_2(). In particular, note that every conjugacy class of commutators in _2() (say, with representative ABA^-1B^-1 of _2() such that A, B ∈_2()) with a given trace T gives rise to an integral solution

    ((A), (B), (AB))

of the Diophantine equation M(x,y,z)=T. The integral solutions to the Markoff equation M(x,y,z)=-2 play an important role in the theory of Diophantine approximation through their relation to extremal numbers via the Lagrange–Markoff spectrum <cit.>, and have been studied recently in <cit.> by applying strong approximation.  Moreover, integral solutions to the Markoff-type equation M(x,y,z)=T for varying T are also of independent number-theoretic interest and have been studied recently in <cit.>. 

The conjugacy classes of _2() with trace T correspond to the _2()-orbits of binary quadratic forms with discriminant T^2-4.  This allows one to use Gauss' reduction theory of indefinite binary quadratic forms, which yields a full set of representatives for the _2()-orbits of binary quadratic forms of any positive discriminant, to obtain the conjugacy classes of _2() with trace T. Furthermore, there is a straightforward condition determining whether each such conjugacy class is in the commutator subgroup [_2(),_2()], since [_2(),_2()] is precisely

    {[ a b; c d ]   (1-c^2)   (bd+3(c-1)d+c+3)+c(a+d-3) ≡ 0 12
     or   (1-   c^2)(bd+3(c+1)d-c+3)+c(a+d+3) ≡ 0 12}

a congruence subgroup of index 6; this follows from the fact that [_2(),_2()] = χ for the surjective homomorphism χ : _2() →/12 defined by 

    [ a b; c d ]↦  (1-c^2)(bd+3(c-1)d+c+3)+c(a+d-3),

as shown in <cit.>. However, there is no known number-theoretic condition determining whether a conjugacy class of _2() with trivial abelianization is in fact a commutator; the only known method to do so is to decompose an element of the conjugacy class into a word in the generators S and T_1, then combinatorially checking whether there exists a cyclic conjugate of this word in the form given by Definition <ref>. For an implementation of this algorithm in , see <cit.>. 

Since every commutator of _2() with trace T gives rise to an integral point on M_T, the natural follow-up question is whether the converse is true. A local version of this holds, in that for every power p^n of a prime p ≥ 5, there exists a commutator ABA^-1B^-1 satisfying (<ref>) p^n <cit.>. The global version of the converse, however, does not hold in general, nor do we know of any number-theoretic condition determining whether an integral point on M_T  comes from a commutator. Then, a question of Sarnak <cit.> asks:


What is the asymptotic number of integral points on M_T that arise from _2()-commutators by the trace identity (<ref>) as |T| →∞?  


On this front, one can consider _2()-commutators as elements of _2()≅/2 * /3, for which Theorem <ref> gives the asymptotic number of conjugacy classes of commutators with a given word length in the generators. Any correlation between the word length of a hyperbolic conjugacy class γ of  _2() and the geometric length of the corresponding geodesic in Γ\ would also yield correlation between the word length of γ and its trace t(γ), defined to be the absolute value of the trace of any representative matrix. This is due to the relation between t(γ) and ℓ(γ) given by

    t(γ) = 2coshℓ(γ)/2,

which is equivalent to the relation (<ref>). Thus, a proof of such a correlation would allow Theorem <ref> to yield information on the number of integral points  on  M_T and M_-T for T > 2  (the condition for hyperbolicity, where we have without loss of generality taken the representative trace  T to be positive)  arising from commutators by (<ref>).





 §.§ Strategy of proof

The main idea behind the proof of Theorem <ref> given in Section <ref> (respectively, the proof of Theorem <ref> given in Section <ref>) is to count the Wicks commutators of F_r (respectively, those of G_1 * G_2) and show that each such Wicks commutator is on average cyclically conjugate to precisely six Wicks commutators (including itself) while keeping the overall error term under control. Thus, we obtain the asymptotic number of conjugacy classes of commutators with length k by dividing the asymptotic number of Wicks commutators of length k by 6. Finally, in Section <ref>, we conclude the paper with a discussion of future  directions for generalization.  




§ PROOF OF THEOREM <REF>



 §.§ Counting the Wicks commutators of F_r

Since the cyclically reduced conjugacy representative of F_r is unique up to cyclic permutation, it suffices to count equivalence classes (with respect to cyclic permutation) of Wicks commutators of length k=2X. Let R_X denote the set of reduced words of length X, of which there are 2r  (2r-1)^X-1. For each such word W, the number of ways to decompose W into A, B, and C (i.e., W=ABC without cancellation) is given by the number of ordered partitions p of X into three (not necessarily nontrivial) parts. 

Define  (W,p) to be a viable pair of length n if W is a length-n reduced word,  p=(n_1,n_2,n_3) is a partition of n into three parts  n_1, n_2, n_3>0, and the  word 

    W' ABCA^-1B^-1C^-1

(obtained by decomposing W as specified by p, in the method described above) is a Wicks commutator. We now show that for a fixed p, the proportion of W ∈ R_X such that (W,p) is viable is given by 

    1/2r + O(1/(2r-1)^n_1-1 +1/(2r-1)^n_2-1 +1/(2r-1)^n_3-1).

It is natural to first count the viable pairs (W,p) of length X. To do so, we will need the following.


In the set of all reduced words of length n beginning with a given letter, the proportion of words ending in any one letter is given by

    1/(2r-1)^n-1·(2r-1)^n-1+O(r)/2r   =  1/2r + O(1/(2r-1)^n-1).



The number of cyclically reduced words with length n is (2r-1)^n+(r-1)(-1)^n+r (see <cit.>). Another way of stating this is that the number a_n of reduced words of length n whose initial  and final letters are inverses (i.e., are not cyclically reduced) is 

    a_n = 2r(2r-1)^n-1-(2r-1)^n   -(r-1)(-1)^n-r
    
       =(2r-1)^n-1-(r-1)(-1)^n-r.

Similarly, let  b_n denote the number of reduced words whose initial  and final letters are equal. Note that {a_n}_n∈_>0 and {b_n}_n∈_>0 are solutions to the recurrence 

    c_n=(2r-1)c_n-2+(2r-2)   (2r(2r-1)^n-3-c_n-2)
    
       =c_n-2+2r(2r-2)(2r-1)^n-3.

Indeed, for n≥ 3, a reduced word whose initial and final letters are inverses is precisely of the form sWs^-1, and there are 2r-1 possible choices for the letter s if W also has the aforementioned property (initial and final letters are inverses) and 2r-2 possible choices if W does not; an analogous discussion holds for the property that the initial and final letters are equal. It follows that the difference {b_n-a_n}_n∈_>0 is a solution to the associated homogeneous recurrence d_n=d_n-2, and thus a linear combination of the functions 1 and n. In light of the initial data a_1=a_2=0 and b_1=b_2=2r, we see that b_n-a_n=2r, which yields

    b_n=(2r-1)^n-1-(r-1)(-1)^n+r.



For s_1,s_2 ∈, let R_n^s_1,s_2⊂ R_n denote the subset of words that begin with s_1 and end with s_2, which gives us a decomposition of R_n into the disjoint union R_n = ⋃_s_1,s_2∈R_n^s_1,s_2. For any s ∈, 

    |R_n^s,s^-1|=a_n/2r=(2r-1)^n-1-(r-1)(-1)^n-r/2r

and 

    |R_n^s,s|=b_n/2r=(2r-1)^n-1-(r-1)(-1)^n+r/2r.

It follows that for t ∈∖{s,s^-1},

    |R_n^s,t| = (2r-1)^n-1-|R_n^s,s^-1|-|R_n^s,s|/2r-2=  1/2r-2( (2r-1)^n-1 -   (2r-1)^n-1-(r-1)(-1)^n/r)
       =(2r-1)^n-1+O(1) /2r,

which completes the proof of our claim.


Now, fix a partition p=(n_1,n_2,n_3) of n into three parts  n_1, n_2, n_3>0, and consider R_X with the uniform probability measure placed on its elements. Within the decomposition W = ABC in accordance with p, let the first letter and last letter of A respectively be a_0 and a_1, and define b_0, b_1, c_0, and c_1 similarly. We will compute the probability that (W,p) is viable for a random W ∈ R_X, i.e., the probability that b_1 ≠ a_0 ^-1, c_0 ≠ a_0, c_1 ≠ a_1, and c_1 ≠ b_0^-1. 

Suppose that a_0=s. Then, by our work above, the set of possible candidates for a_1b_0 is the 2r(2r-1)-element set S={wz : w, z ∈ , w ≠ z^-1}, each of which has probability 

    1/2r-1(1/2r+O(1/(2r-1)^n_1-1)) = 1/2r(2r-1)+O(1/(2r-1)^n_1).

We fix a choice of a_1b_0 in S, and all probabilities from now on are conditional on this event. The set of possible candidates for b_1c_0 is also S, each of which has probability 1/2r(2r-1)+O(1/(2r-1)^n_2). Let S' ⊂ S be the subset of  possible candidates for b_1c_0 that satisfy the  conditions a_0 ≠ b_1^-1 and a_0 ≠ c_0. The cardinality of S' can be computed as follows: there are 2r-1 choices for b_1 satisfying s ≠ b_1^-1, and conditional on this, there are 2r-2 choices for c_0 satisfying s ≠ c_0 and b_0 ≠ c_0^-1, for a total of (2r-1)(2r-2) elements of S'. Fix a choice of b_1c_0 in S', and all probabilities from now on are conditional on this event. Since a_1 ≠  b_0^-1, the conditions  c_1 ≠ a_1 and c_1 ≠ b_0^-1 leave precisely 2r-2 (out of 2r) possible values for c_1, so the probability that c_1 satisfies these conditions is 

    (2r-2) (1/2r + O(1/(2r-1)^n_3-1) ).

 Overall, we have that the probability that (W,p) is viable is

    2r(2r-1)   (1/2r(2r-1)   + O(1/(2r-1)^n_1) )
    
       · (2r-1)(2r-2)  (1/2r(2r-1) + O(1/(2r-1)^n_2) ) 
       · (2r-2) (1/2r + O(1/(2r-1)^n_3-1) )
     =
    (2r-2/2r)^2       (1+ O(1/(2r-1)^n_1-2) )   (1+ O(1/(2r-1)^n_2-2) ) 
       · (1+ O(1/(2r-1)^n_3-2) )
     
     =(2r-2/2r)^2    (1+ O(1/(2r-1)^n_1-2+1/(2r-1)^n_2-2+1/(2r-1)^n_3-2) ).

Thus, the number of viable pairs (W,p) of length X is  given by

    ∑_n_1+n_2+n_3 = X   2r (2r-1)^X-1(2r-2/2r)^2 
       · (1+ O(1/(2r-1)^n_1-2+1/(2r-1)^n_2-2+1/(2r-1)^n_3-2) )
    
       =(2r-2)^2(2r-1)^X-1/2r( (X-2)(X-1)/2
       +∑_n_1+n_2+n_3 = X O(1/(2r-1)^n_1-2 +1/(2r-1)^n_2-2 +1/(2r-1)^n_3-2)  ) 
       =(2r-2)^2(2r-1)^X-1/2r((X-2)(X-1)/2 + 3 · O(∑_n_1+n_2+n_3 = X1/(2r-1)^n_1-2) )
      
       = (2r-2)^2(2r-1)^X-1/2r((X-2)(X-1)/2
       + O(∑_n_1=1^X-2  (X-1-n_1)1/(2r-1)^n_1-2) )
       
       = (2r-2)^2(2r-1)^X-1/2r((X-2)(X-1)/2
       + O((2r-1)^2((2r-1)^2-X+X(2r-2)-4r+3)/(2r-2)^2) )
      
       =(2r-2)^2(2r-1)^X-1/4r(X^2 + O( r^2/(2r-1)^X+rX)  ),

where all sums with n_1+n_2+n_3 = X are taken over integers n_1,n_2,n_3>0.

There could a priori be a Wicks commutator ABCA^-1B^-1C^-1 arising from distinct viable pairs, say (W, p_1) and (W, p_2) for p_1=(n_1,n_2,n_3), and p_2 = (m_1,m_2,m_3). We show that the number of such commutators is small. 

Label the letters of A as a_1, …, a_n_1, and label the letters of B, C, D, E, and F similarly. We have that w_1, w_2, and w_3 as subwords comprised of the letters 

    d_ℓ^-1, …, d_1^-1, a_1, …, a_n_1, b_1, …, b_n_2,  d_1, …, d_ℓ, e_1, …, e_n_3-2ℓ.

Then, note that the second half of W' can be considered in two forms:

    F  A^-1 B^-1  F^-1E^-1= w_1^-1 w_2^-1 w_3^-1 .

Equivalently, this equality can be written as 

    EF B  A F^-1 = w_3 w_2  w_1,

where we reiterate that we substitute the letters of (<ref>), in the correct order, for w_3, w_2, and w_1.

Consider the function f mapping the ordered set of symbols of the left-hand side of (<ref>),

    {e_1,…,e_n_3-2ℓ, f_1,…,f_ℓ,b_1,…,b_n_2, a_1,…,a_n_1, f_ℓ^-1,…,f_1^-1},

to the ordered set  of symbols of the right-hand side of (<ref>), defined so that it maps the ith leftmost letter of the left-hand side to the ith leftmost letter of the right-hand side. 



Let W=ABC be its decomposition with respect to p_1, and W=A'B'C' its decomposition with respect to p_2.  Consider the function f : {1, …, X}→{1, …, X}^2 that maps i to (j,k) in the following way: the ith letter of A^-1B^-1C^-1=A^' -1B^' -1C^' -1, when viewed in terms of the decomposition given by p_1, is the inverse of the jth letter of W, and  when viewed in terms of the decomposition given by p_2, is the inverse of the kth letter of W. For example, the first letter of A^-1B^-1C^-1 is defined to be the inverse of the n_1th letter when the decomposition is in terms of p_1, and is defined to be the m_1th letter of W when in terms of p_2, so f(1) = (n_1,m_1). We consider two cases: when the two entries of f(i) are distinct for all i, and otherwise. In the first case, the following algorithm allows us to reduce the degrees of freedom for the letters of W by at least half.


  * Let i=1. For f(i)=(j_i,k_i), do the following:


  * If neither the j_ith or the k_ith position has an indeterminate variable assigned to it, then assign a new indeterminate variable simultaneously to the j_ith and k_ith positions. This increases the number of indeterminate variables by two.


  * If just one of the j_ith and the k_ith positions has an indeterminate variable assigned to it, but the other does not, then assign this indeterminate variable to the former. 


  * If both the j_ith and the k_ith positions have the same indeterminate variable assigned to them, then make no changes. 


  * If the j_ith and the k_ith positions have distinct indeterminate variables assigned to them, then set these indeterminate variables equal to each other. This decreases the number of indeterminate variables by one.


  * Incrementing i by one each time,  repeat this procedure for all 1 ≤ i ≤ X. 

By our hypothesis that the two entries of f(i) are distinct for all i, the number of indeterminate variables, which precisely represents the number of degrees of freedom for the word W such that (W,p_1) and (W,p_2) give rise to the same commutator, is ≤ X/2. It follows that there are only O((2r-1)^X/2) of such words for each pair p_1, p_2. 

Now, consider the next case that there exists an i such that the two entries of f(i) are equal. Then, we consider the following three cases for W, p_1, and p_2.

Case 1. Suppose the smallest i such that the two entries of f(i) are equal satisfies that this entry is a position in A. Then, n_1=m_1 must be this entry and i must equal 1, since otherwise we can continue to decrement i so that the two entries of f(i) are incremented and remain equal, a contradiction. Next, the subwords B^-1C^-1 and B'^-1C'^-1 must be equal. Without loss of generality, suppose that n_2 > m_2. Decompose B=B'D so that our condition B^-1C^-1=B'^-1C'^-1 is precisely CB'D = DCB'. Since CB' and D commute, we have that they are both powers of a common subword V. Since n_2>m_2, we have that D is a positive power, and likewise CB' is a positive power 
both of them must be positive powers, since otherwise there would be cancellation, which contradicts that the commutator is cyclically reduced. We can bound the number of (W,p) satisfying this case by counting, for each i=X-n_1  and for each  divisor d | i (denoting the length of V), the number of ways to divide the right subword of length i into two subwords B and C, and the number of degrees of freedom. Thus, the number of double-countable Wicks commutators in  this case can be bounded from above by

    ∑_i=1^X-1∑_d | i (i-d+1)    (2r-1)^X-i+d
       =
    (X-1)(2r-1)^X +∑_i=1^X-1∑_d | i,  d ≠ i  (i-d+1) (2r-1)^X-i+d
       ≤  X(2r-1)^X +(2r-1)^X ∑_i=2^X-1∑_1≤ d ≤i/2 (i-d+1)  (2r-1)^-i+d
       =  X(2r-1)^X +(2r-1)^X  ∑_i=2^X-1 (2r-1)^-i∑_1≤ d ≤i/2 (i-d+1)  (2r-1)^d
       ≤ X(2r-1)^X+ (2r-1)^X ∑_i=2^X-1 (2r-1)^-i+1
       · i (2r-2)( (2r-1)^i/2 -2) +2(2r-1) ( (2r-1)^i/2 -1 )/2(2r-2)^2
       ≪ X(2r-1)^X ,

which is dominated by our error term. 


   =  (2r-1)^X ·1/2(2r-2)^3(2r-1)^X ( (X-2)(2r-1)^X/2 +2 - (X+2)(2r-1)^X/2 +3 

     -(2r-1)^X+1/2 - 6(2r-1)^X+1 + 5(2r-1)^X+3/2 +4(2r-1)^X+2 + (X-1)(2r-1)^X+3/2 

   -(X+3)(2r-1)^X+5/2 +2(2r-1)^X + 2(X+1)(2r-1)^3 -2(X-1)(2r-1)^2), 

Case 2. Suppose the smallest i such that the two entries of f(i) are equal satisfies that this entry is a position in C. An argument symmetric to the one above can be given to show that the above expression is also an upper bound for the number of  (W,p) satisfying this case. 


Case 3. Suppose the smallest i such that the two entries of f(i) are equal satisfies that this entry is a position in B. Without loss of generality, suppose n_1>m_1. Then, f(m_1+1)=(n_1-m_1,m_1+m_2), and by an argument similar to that in Case (1), we have that the simultaneous entry of the aforementioned f(i) must be n_1+n_2, with n_1-m_1=n_3-m_3 so that f(m_1+1)=(n_1-m_1,n_1+n_2+(n_1-m_1)). Thus, divide W into DEFGH so that |D|+|E| = n_1, |G|+|H|=n_3, and |E|=|G|. Then, (W,p_1) gives rise to the commutator WE^-1D^-1F^-1H^-1G^-1, while (W,p_2) gives rise to the commutator WD^-1G^-1F^-1E^-1H^-1. Since these are equal, it follows that DE =GD and GH=HE. Note that if a word □ satisfies the equality □  E= G □ without cancellation, then □ is uniquely determined, since one can inductively identify the letters of □ from left to right (or right to left). It follows that D=H. But this contradicts the assumption that DEFGHD^-1G^-1F^-1E^-1H^-1 is cyclically reduced.

Note that pairs (W,p) such that n_i=0 for some i ∈{1,2,3} are counted in the above cases. This justifies our assumption of n_1,n_2,n_3 >0 in our earlier counting of the main term by passing to the task of counting viable pairs of length X, which a priori only accounts for Wicks commutators ABCA^-1B^-1C^-1 such that |A|,|B|,|C|>0. We have thus shown the following.

The total number of Wicks commutators with length 2X is given by

    (2r-2)^2(2r-1)^X-1/4r(X^2 + O_r( X)  ).





 §.§ A conjugacy class of commutators  contains six Wicks commutators on average

We need to count the number of  conjugacy classes containing at least one Wicks commutator. Consider the conjugacy class  of the Wicks commutator W'=ABCA^-1B^-1C^-1 arising from (W,p), where p=(n_1,n_2,n_3) with n_1,n_2,n_3>0; recall that the Wicks commutator not satisfying this latter hypothesis are negligible. Note that the minimum-length elements in a conjugacy class are precisely the cyclically reduced words, and that two cyclically reduced words are conjugate if and only if they are cyclically conjugate. The Wicks commutators 

    BCA^-1B^-1C^-1A, CA^-1B^-1C^-1AB,    A^-1B^-1C^-1ABC, 
    
       B^-1C^-1ABCA^-1,  and  C^-1ABCA^-1B^-1

are conjugates of W'. We show that the number of other Wicks commutators in  is on average negligible.

For an arbitrary 1 ≤ℓ≤ n_3/2 denoting the number of letters of the conjugation,  let C=DEF be a decomposition without cancellation such that |D|=|F|= ℓ. Label the letters of W' by A=a_1 ⋯ a_n_1, B=b_1 ⋯ b_n_2, D=d_1 ⋯ d_ℓ, E=e_1⋯ e_n_3-2ℓ, and F =f_1⋯ f_ℓ.  Consider the cyclic conjugate 

    W” D^-1ABDEFA^-1B^-1F^-1E^-1

of W'. We wish to show that on average, W” is not a Wicks commutator. Suppose the  contrary, that there exists a partition p'=(m_1,m_2,m_3) of X into three parts such that

    W”=D^-1ABDEFA^-1B^-1F^-1E^-1= w_1w_2w_3w_1^-1w_2^-1w_3^-1

for subwords w_1, w_2, and w_3 of lengths m_1, m_2, and m_3. 




Label the letters of A from left to right as a_1, …, a_n_1, and label the letters of B,C,D,E, and F similarly. We have that w_1, w_2, and w_3 are subwords comprised of the letters 

    d_ℓ^-1, …, d_1^-1, a_1, …, a_n_1, b_1, …, b_n_2, d_1, …, d_ℓ, e_1, …, e_n_3-2ℓ.

Then, note that the second half of W' can be considered in two forms:

    F A^-1  B^-1  F^-1E^-1= w_1^-1  w_2^-1  w_3^-1 .

Equivalently, this equality can be written as 

    EF B A F^-1 = w_3 w_2 w_1,

where we reiterate that we substitute the letters of (<ref>), in the correct order, for w_3, w_2, and w_1.

Consider the function g mapping the ordered set of symbols of the left-hand side,

    {e_1,…,e_n_3-2ℓ, f_1,…,f_ℓ,
    b_1,…,b_n_2,a_1,…,a_n_1, f_ℓ^-1,…,f_1^-1},

to the  ordered set  of symbols of the right-hand side of (<ref>), which are just the symbols of (<ref>) reordered appropriately. Specifically, g  maps the ith leftmost letter of the left-hand side of (<ref>) to the ith leftmost letter of the right-hand side. 

First, suppose g has no fixed points (𝔦 such that g(𝔦)=). Then, use an algorithm similar to the previous one to conclude that there are ≤ X/2 degrees of freedom for ABDEF, so W must be one of only O((2r-1)^X/2) choices (for each choice of ℓ and p').

Now, suppose that there exists an 𝔦 such that g(𝔦)=. Such fixed points  must be letters of A, B, or E. We first consider the case that all the fixed points are letters of only one of A, B, and E. In this case, we consider the following subcases for W, p, and p': 

Case 1. Suppose that the fixed points are letters of E. Then, all of the fixed points must be in one of w_2 and w_3; they cannot be in w_1, since this would mean that w_1 contains e_1, but e_1 is then necessarily located at different positions in the left-hand side and right-hand side of (<ref>). Suppose that the fixed points of E are in w_3. Then, in order for the letters of E to match, we require that w_3=E. This means that g(f_1) is the first letter of w_2, which is adjacent to the last letter of w_1. But the last letter of w_1 is g(f_1^-1), which shows that we have adjacent letters that are inverses. This contradicts the fact that W' is  cyclically reduced. 

Next, suppose all the fixed points are in w_2. Then, we  must have that m_3=n_3-2ℓ-(m_2+m_3) so that the first letter of w_2 is at the same position in both the left-hand and right-hand side. Thus, m_2+2m_3=n_3-2ℓ, which means there are ≤ (n_3-2ℓ)/2 choices for p' parametrized by m_3 ≤(n_3-2ℓ)/2 . For each such choice of p', there are m_2=n_3-2ℓ-2m_3 fixed letters,  from e_m_3+1 to e_n_3-2ℓ-m_3, and (X-(n_3-2ℓ-2m_3))/2 non-fixed letters. Counting across all choices of values for the letters, p, p', and ℓ,  we have that the number of additional Wicks commutators arising from this case is bounded from above by

    ∑_n_1=0^X∑_n_2=0^X-n_1∑_ℓ=1^X-n_1-n_2/2∑_m_3=0^n_3-2ℓ/2  (2r-1)^X+n_3-2ℓ-2m_3/2≪ (2r-1)^X,

which is dominated by our error term.


Case 2.  Suppose that the fixed points are letters of B. Then, all of the fixed points must be in one of w_1, w_2, and w_3. First, suppose they are in w_1. Then, note that g(f_ℓ)=a_n_1, but we also have g(a_n_1) is next to g(f_ℓ^-1), which leads to the contradiction that a letter cannot equal its inverse. Second, suppose the fixed letters are in w_3. Then, g(d_1)=a_1, but a_1 is adjacent to d_1^-1, a contradiction. 

Thus, the fixed points of B must be in w_2. We consider three subcases: n_2 > m_2, n_2< m_2, and n_2=m_2. If n_2 > m_2, then  in order for the letters of B to match, we require that the leftmost fixed letter of B is b_n_2-m_2/2+1.  But then b_n_2-m_2/2 is both equal to f_1^-1 (since g(f_1^-1)=b_n_2-m_2/2) and e_n_3-2ℓ (since g(b_n_2-m_2/2)=e_n_3-2ℓ) as letters of , which contradicts the fact that f_1 and e_n_3-2ℓ are adjacent. If n_2 < m_2, then g(f_ℓ)=a_n_1, but also g(a_n_1) is the letter in D^-1A that is left of the letter g(f^-1_ℓ), giving us the contradiction that the letter in  in the position f_ℓ is adjacent to the letter in the position f_ℓ^-1. This implies that n_2=m_2, from which we can use an argument similar to that in Case 1 of the previous casework (showing that W' on average can be only decomposed as a commutator in one way) to conclude that A is a power of D^-1 and E, a power of D. It follows that our original (W,p) is one of the pairs falling under Case 1 of the previous casework, which are negligible. 

Case 3. Suppose that the fixed points are letters of A. Then, all of the fixed points must be in one of w_1 and w_2; they cannot be in w_3, since then there must be more than ℓ letters right of A. Suppose the fixed letters of A are in w_1. Then, we must have g(b_n_2)=d_1^-1, which contradicts the fact that b_n_2 is adjacent to d_1.  Therefore,  the fixed points are necessarily in w_2. This requires that m_1-ℓ=n_1+ℓ-(m_1-m_2) in order for the letters of A to be in matching positions. Thus, we have m_2=n_1+2ℓ-2m_1.  Note then that p' is parametrized by m_1 ≤ (n_1+2ℓ)/2. For each choice of p', we have n_1 fixed letters (and (X+n_1)/2 ≤ (X+n_1-m_1+ℓ)/2 overall degrees of freedom) if m_1 ≤ℓ, and  n_1-(m_1-ℓ) fixed letters (and (X+n_1-m_1+ℓ)/2 overall degrees of freedom) if m_1 > ℓ. Thus, counting across all choices of values for the letters, p, p', and ℓ,  we have that the number of additional Wicks commutators arising from this case is bounded from above by

    ∑_n_1=0^X∑_n_2=0^X-n_1∑_ℓ=1^X-n_1-n_2/2∑_m_1=0^n_1+2ℓ/2 (2r-1)^X+n_1+ℓ-m_1/2≪ (2r-1)^X,

which is dominated by our error term.

Next, consider the case where the fixed points are in two of A, B, and E. It is necessary that the fixed letters inside these two subwords must respectively be in two distinct subwords among w_1,w_2, and w_3. However, we have shown above that the subwords w_1 and w_3 cannot contain fixed points, a contradiction. Finally, the fixed letters cannot be in all of A, B, and E. Indeed, if this were true, then in order for the letters of A and E to match, we require m_1=n_1+2ℓ and m_3=n_3. But then the letters of B cannot possibly match, a contradiction. 

If ℓ≥ n_3/2, then we can think of our commutator as a cyclic conjugate of C^-1ABCA^-1B^-1 such that the letters are moved from left to right. A symmetric argument like above gives us the same conclusion for this case. We have thus shown that the number of conjugacy classes of commutators with length 2X is given by

    1/6·(2r-2)^2(2r-1)^X-1/4r(X^2 + O_r(X)  )= (2r-2)^2(2r-1)^X-1/24r(X^2 + O_r( X)  ),

as needed.




§ PROOF OF THEOREM <REF>



 §.§ Counting the Wicks commutators of G_1 * G_2

In addition to his theorem classifying commutators of free groups, Wicks <cit.> also proved the following analogous theorem characterizing all commutators of a free product of arbitrary groups.

A word in _i∈ I G_i is a commutator if and only if it is a conjugate of one of the following fully cyclically reduced products:


  * a word comprised of a single letter that is a commutator in its factor G_i,

  * X_1 X _2^-1, where X is nontrivial and _1, _2 belong to the same factor G_i as conjugate elements,

  * X _1 Y _2 X^-1_3 Y^-1_4, where X and Y are both nontrivial, _1, _2, _3,_4 belong to the same factor G_i, and _4_3_2_1 is trivial,

  * XYZX^-1Y^-1Z^-1,

  * X  Y _1  Z  X^-1_2 Y^-1  Z^-1_3, where Y and at least one of X and Z is nontrivial,  _1, _2, _3 belong to the same factor G_i, and _3_2_1 is trivial,

  * X _1 Y _1  Z _2 X^-1_2 Y^-1_3 Z^-1_3, where _1, _2, _3 belong to the same factor G_i and _1, _2,_3, to G_j, _3_2_1=_3_2_1=1, and either _1, _2, _3, _1, _2,_3 are not all in the same factor or X,Y,Z are all nontrivial.


Note that in the above, the Greek letters are assumed to be nontrivial. This convention  is used later in our proof, where when a Greek letter  is said to satisfy ∈ G_i, we mean that  is a nontrivial element of G_i. 

Equipped with a complete classification of commutators in an arbitrary free product, we proceed with our proof of Theorem <ref>. Let G_1 * G_2 be the free product of two nontrivial finite groups G_1 and G_2. The letters of a cyclically reduced word of G must alternate between equal numbers of elements of  and elements of , and thus must be of even length. Let C be a fully reduced commutator of G_1 * G_2. If C is of the form (d) listed in Theorem <ref> with none of X,Y, and Z trivial, then the last letter of X is in different factors compared to the first letter of Y and the last letter of Z, which must also be in different factors, a contradiction. If C is of the form (e), then _1, _2, and _3 must be in the same free-product factor, but this would imply that the last letter of Y is in different free factors compared to the first letter of Z  and the first letter of X; this contradicts the similar implication that the first letter of Z  and the first letter of X are in different factors. Thus, C must be of the form (a), (b), (c), (f), or XYX^-1Y^-1 (i.e., of the form (d) with |Z|=0). However, if C is of the form (f), then G_i and G_j must be the same free-product factor, since otherwise _1 would be adjacent to letters of distinct free-product factors, a contradiction.

By Wicks' theorem for free products, we need to count cyclic conjugacy classes of Wicks commutators of G_1 * G_2. It follows from the discussion above that every commutator of W is conjugate to one of the fully cyclically reduced forms listed under Definition <ref>. These general forms, labeled from (1) to (9), have without loss of generality been taken to have the letters of odd position be in G_1 and those of even position in G_2. Throughout this proof, we will regularly use the terminology Wicks commutators of the form (i), where 1 ≤ i ≤ 9, to refer to the corresponding general form listed under Definition <ref>.

Consider Wicks commutators of G_1 * G_2 with length k, where k is a multiple of 4. Let X=k/4, so that the left-half subword of W contains X letters of the G_1 factor and X letters of the G_2 factor, which are placed in  alternating order.  The commutators of the forms (1) and (2) are O(1) in number and all have length 1. The commutators of the forms (3) and (4) are O(^X^X) in number, since there are ≤ X degrees of freedom among the letters of each free-product factor. The commutators of the forms (5), (6), and (7) are O(X  ^X^X) in number, since there are ≤ X degrees of freedom for the X letters of each free-product factor and O(X) possible pairs of values for |A| and |B|, by an argument similar to that used in Section <ref>.


The number of Wicks commutators of the form (8) is

    (X-5)(X-4)/2( |G_2|-2 )^2  ^X ^X-1.

Indeed, we have ( |G_2|-2 ) distinct choices of the triple (_1, _2, _3), since there are |G_2|-1  choices for _1, |G_2|-2 choices for _2 ≠_1^-1, and _3 is uniquely determined from the previous choices.  Likewise, there are ( |G_2|-2 ) distinct choices of the triple (_1, _2, _3). Finally, there are (X-5)(X-4)/2 partitions of X-3 into three nontrivial parts (n_1,n_2,n_3) such that  |A| = 2n_1 +1, |B|= 2n_2 +1, and |C|= 2n_3 +1; X degrees of freedom for choosing the G_1-letters of A, B, and C; and X-3 degrees of freedom for choosing the G_2-letters of A, B, and C. By an analogous argument, the number of Wicks commutators of the form (9) is 

    (X-5)(X-4)/2( |G_1|-2 )^2   ^X-1^X.

We define a generic Wicks commutator of G_1 * G_2 to be one of the form (8) or (9). These comprise the main term of the total number of Wicks commutators of length k=4X, since we have seen above that the non-generic Wicks commutators, those of the forms (1)-(7), comprise a negligible subset. Overall, we have shown the following.

The total number of Wicks commutators with length 4X is given by

    1/2(( |G_2|-2 )^2 +( |G_1|-2 )^2 )     X^2  ^X-1^X-1
     +   O( X^X^X).





 §.§ A conjugacy class of commutators contains six Wicks commutators on average
 We need to count the number of conjugacy classes containing at least one generic Wicks commutator. As before, let  be the conjugacy class of the Wicks commutator W A_1 B_1 C_2 A^-1_2 B^-1_3 C^-1_3, with (n_1,n_2,n_3) a partition of X-3 and |A|=2n_1+1, |B|=2n_2+1, and |C|=2n_3+1. We wish to show that on average,  does not contain generic Wicks commutators other than the six obvious ones:  

    W, B_1 C_2 A^-1_2 B^-1_3 C^-1   _3 A _1, C_2 A^-1_2 B^-1_3 C^-1_3 A _1 B _1, 
    
       A^-1_2 B^-1_3 C^-1_3 A _1 B _1 C_2, B^-1_3 C^-1_3 A_1B_1 C_2 A^-1_2,
    
       and  C^-1_3 A_1 B _1 C _2A^-1_2 B^-1_3.

We wish to show that any other cyclic conjugate W' of W is not on average a Wicks commutator. 

Suppose the contrary. One of the ways this can happen is if there is such a W' that is of the form (8), i.e, there exists a partition p'=(m_1,m_2,m_3) of X-3 into three parts such that 

    W'=w_1'_1 w_2'_1 w_3'_2 w_1^-1'_2 w_2^-1'_3 w_3^-1'_3

 for subwords w_1, w_2, and  w_3 of lengths 2m_1+1, 2m_2+1, and 2m_3+1 given by p', and 
 
    '_1,'_2,'_3, '_1,'_2, '_3 ∈ G_2.

 We will see that the argument showing that these exceptions are negligible also shows that the exceptions such that W' is of the other Wicks-commutator forms are also negligible.

Suppose the number of letters of the conjugation is 2ℓ, where ℓ≤ (n_3+1)/2 is arbitrary. For the desired uniformity of our presented argument, we assume that ℓ >0, although the exceptions in the ℓ=0 case can be bounded similarly. We decompose C=DEF  without cancellation so that |D|=|F|= 2ℓ-1. Label the letters of W by A=a_1 a̅_1a_2 a̅_2 ⋯a̅_n_1 a_n_1+1, B=b_1 b̅_1b_2 b̅_2⋯b̅_n_2 b_n_2+1, D=d_1d̅_1 ⋯d̅_ℓ-1 d_ℓ, E=e̅_1 e_2 e̅_2  ⋯   e_n_3-2ℓ +2e̅_n_3-2ℓ+2, and F =f_1 f̅_1 ⋯f̅_ℓ-1 f_ℓ; note that the barred letters denote the G_2-letters and the non-barred letters, the G_1-letters. Consider the cyclic conjugate 

    W' D^-1_3 A_1 B_1 D E F_2 A^-1_2 B^-1_3 F^-1E^-1

of W. We wish to show that on average, W' is not a Wicks commutator.

The subwords w_1, w_2, and w_3 are  comprised of the letters 

    d_ℓ^-1, …, d_1^-1, β_3, a_1, …, a_n_1+1, α_1, b_1, …, b_n_2+1, _1, d_1, …, d_ℓ, e̅_1, …, e̅_n_3-2ℓ+2,

except three of these letters instead correspond to '_1,'_2, and '_3 and thus omitted. Then, note that the second half of W' can be considered in two forms:

    F _2 A^-1_3 B^-1_3 F^-1E^-1= w_1^-1'_2 w_2^-1'_3 w_3^-1'_3.

Equivalently, this equality can be written as 

    EF_3^-1 B _3^-1 A _2^-1 F^-1 = ^' -1_3w_3^' -1_3 w_2 ^' -1_2 w_1,

where we reiterate that we substitute the appropriate letters of (<ref>), in the correct order, for w_3, w_2, and w_1.

Consider the function g mapping the ordered set of symbols of the left-hand side of (<ref>),

    {e̅_1,…,e̅_n_3-2ℓ+2, f_1,…, f_ℓ,_3^-1, b_1,…,b_n_2+1, _3^-1,a_1,…,a_n_1+1,_2^-1 ,f_ℓ^-1,…,f_1^-1},

to the ordered  set  of symbols of the right-hand side of (<ref>), which when reordered are comprised of the letters of (<ref>) except we replace the (2m_1+2)th, (2m_1+2m_2+4)th, and (2m_1+2m_2+2m_3+6)th leftmost letters in (<ref>) with ^' -1_2,^' -1_3, and ^' -1_3; note that the  (2m_1+2m_2+2m_3+6)th letter is always e̅_n_3-2ℓ+2. Specifically, g  maps the ith leftmost letter of the left-hand side of (<ref>) to the ith leftmost letter of the right-hand side. 

First, suppose g has no fixed points (𝔦 such that g(𝔦)=). Then, use an algorithm similar to the one used in Section <ref> to conclude that there are ≤ X/2 degrees of freedom for the G_1-letters of ABDEF, and likewise for the G_2-letters. So, W must be one of only O(^X/2^X/2) choices (for each choice of ℓ and p').

Now, suppose that there exists an 𝔦 such that g(𝔦)=. Such fixed points  must be letters of A, B, or E. We first consider the case that all the fixed points are letters of only one of A, B, and E. In this case, we consider the following subcases for W, p, and p': 

Case 1. Suppose that the fixed points are letters of E. Then, all of the fixed points must be in one of w_2 and w_3; they cannot be in w_1 since this would mean that w_1 contains e_1, but e_1 is necessarily located at different positions in the left-hand side and right-hand side of (<ref>). If all the fixed points are in w_2, then we require that 

    1+(2m_3+1)+1=(2n_3-4ℓ+3)-((2m_2+1)+1+(2m_3+1)+1),

 in order for the first letter of w_2 to be at the same position in both the left-hand and right-hand side. Hence, we have m_2+2m_3=n_3-2ℓ-2, which means there are ≤ (n_3-2ℓ-2)/2 choices for p' parametrized by m_3 ≤(n_3-2ℓ-2)/2 . For each such choice of p', there are 
 
    2n_3-4ℓ+3 -2((2m_3+1)+1+1)=2n_3-4ℓ-4m_3-3

fixed letters excluding the first 2m_3+3 and the last 2m_3+3 letters of E, with n_3-2ℓ-2m_3-1 of them in G_1 and n_3-2ℓ-2m_3-2 in G_2. There are ≤(X-(n_3-2ℓ-2m_3-1))/2 degrees of freedom for the non-fixed letters in G_1 and ≤(X-(n_3-2ℓ-2m_3-2))/2  degrees of freedom for the fixed letters in G_2. Thus, counting across all choices of values for the letters, p, p', and ℓ,  we have that the number of additional Wicks commutators arising from this case is bounded from above by

    ∑_n_1=0^X-3∑_n_2=0^X-3-n_1∑_ℓ=1^X-2-n_1-n_2/2∑_m_3=0^n_3-2ℓ-2 /2^X+n_3-2ℓ-2m_3-1/2   ^X+n_3-2ℓ-2m_3-2/2
       ≪^X^X,

which is dominated by our error term.

 Next, suppose the fixed letters are in w_3.  Then, it is necessary that g(e̅_1)=^' -1_3 and w_3 is given by E with the first and last letters omitted. Thus, we have that m_3=n_3-2ℓ, so the number of possible choices for p' is at most the number of partitions of X-3-n_3+2ℓ into two nontrivial parts, which is X-3-n_3+2ℓ. For each choice of p', there are n_3-2ℓ+1 fixed letters in G_1 and n_3-2ℓ fixed letters in G_2. Additionally, there are ≤(X-(n_3-2ℓ+1))/2  degrees of freedom for the non-fixed letters in G_1 and ≤(X-(n_3-2ℓ))/2  degrees of freedom for the  non-fixed letters in G_2. Counting across all choices of values for the letters, p, p', and ℓ,  we have that the number of additional Wicks commutators arising from this case is bounded from above by

    ∑_n_1=0^X-3∑_n_2=0^X-3-n_1∑_ℓ=1^X-2-n_1-n_2/2 (X-3-n_3+2ℓ)  ^X-n_3+2ℓ-1/2   ^X-n_3+2ℓ/2
       ≪^X^X,

which is dominated by our error term.

Case 2.  Suppose that the fixed points are letters of B. Then, all of the fixed points must be in one of w_1, w_2, and w_3. First, suppose they are in w_1. This requires that w_1=D^-1_3 A_1 B _1 V, where V is the  left subword of D E having length 2n_1+2ℓ+1 (the length of A_2^-1F^-1). All letters of B are thus included in w_1.   We have 

    2m_1+1=(2ℓ-1)+1+(2n_1+1)+1+(2n_2+1)   +1+(2n_1+2ℓ+1)
       =4n_1+2n_2+4ℓ+5,

so m_1 = 2n_1+n_2+2ℓ+2. Thus, the number of possible choices for p' is at most the number of partitions of X-3-2n_1-n_2-2ℓ-2 into two nontrivial parts, which is X-5-2n_1-n_2-2ℓ≤ X-3-n_1-n_2-2ℓ (the latter is guaranteed to be nonnegative for any choice of p). The fixed letters are precisely the letters of B, so for each choice of p', we have n_2+1 fixed G_1-letters, n_2 fixed G_2-letters, ≤ (X-n_2-1)/2  degrees of freedom for the non-fixed G_1-letters, and ≤ (X-n_2)/2  degrees of freedom for the non-fixed G_2-letters. Counting across all choices of values for the letters, p, p', and ℓ,  we have that the number of additional Wicks commutators arising from this case is bounded from above by

    ∑_n_1=0^X-3∑_n_2=0^X-3-n_1∑_ℓ=1^X-2-n_1-n_2/2 (X-3-n_1-n_2-2ℓ) ^X+n_2+1/2   ^X+n_2/2
       ≪^X^X,

which is dominated by our error term.

Next, suppose that the fixed points are in w_2. Then, we require that the difference between the lengths of  EF_3^-1 (length 2n_3-2ℓ+3) and _3^-1A_2^-1F^-1 (length 2n_1+2ℓ+2) is the same as that between ^'-1_3w_3_3^'-1 (length 2m_3+3) and ^'-1_2w_1 (length 2m_1+2). Furthermore, the number of (fixed) letters of B in w_2 is 2n_2+1 if 

    j 2n_1+2ℓ+2-(2m_1+2)/2 = 2n_3-2ℓ+3-(2m_3+3)/2
 
is negative and 2(n_2-j)+1 if j ≥ 0. First, suppose that j ≥ 0. In this  case, p' is determined by the choice of j ≤ n_2/2, for which there are n_2-2j+1 fixed G_1-letters and n_2-2j fixed G_2-letters of B. There are ≤(X-(n_2-2j+1))/2  degrees of freedom for the non-fixed letters of G_1 and ≤ (X-n_2+2j)/2 degrees of freedom for the non-fixed letters of G_2, so overall, we can count across all choices of values for the letters, p, j, and ℓ to get that the number of additional Wicks commutators arising from this case is bounded from above by

    ∑_n_1=0^X-3∑_n_3=0^X-3-n_1∑_ℓ=1^n_3/2∑_j=0^X-3-n_1-n_3/2^X+n_2-2j+1/2   ^X+n_2-2j/2
       ≪^X^X,

which is dominated by our error term.

Now, suppose that j <0. In this case, -j=m_1-n_1-ℓ=m_3-n_3+ℓ is a positive integer satisfying (n_1+ℓ-j) +(n_3-ℓ-j) = m_1+m_3 ≤ X-3, which gives the condition -j ≤ (X-3-n_1-n_3)/2 = n_2/2. Note that p' is determined by the choice of -j, for which there are n_2+1 fixed G_1-letters and n_2 fixed G_2-letters, all of which are in B. The non-fixed letters in G_1 have ≤ (X-n_2-1)/2 degrees of freedom and those in G_2 have ≤ (X-n_2)/2 degrees of freedom. Overall, we can count across all choices of values for the letters, p, -j, and ℓ to get   that the number of additional Wicks commutators arising from this case is bounded from above by

    ∑_n_1=0^X-3∑_n_2=0^X-3-n_1∑_ℓ=1^X-2-n_1-n_2/2∑_-j=1^n_2/2^X+n_2-2j+1/2   ^X+n_2-2j/2
    ≪   ^X^X,

which is dominated by our error term.

Finally, suppose that the fixed points are in w_3.  This requires that w_3 is given by V B _1 DE with the last letter omitted, where V is the right subword of D^-1_3 A_1 having length (2n_3-4ℓ+3)+(2ℓ-1)=2n_3-2ℓ+2, one less than that of EF_3^-1. All letters of B are thus included in w_3. We have that
 
    2m_3+1=(2n_3-2ℓ+2)+(2n_2+1)+1+(2ℓ-1)   +(2n_3-4ℓ+2)
       =2n_2+4n_3 - 4ℓ+5,

so m_3=n_2+2n_3-2ℓ+2. Thus, the number of possible choices for p' is at most the number of partitions of X-5-n_2-2n_3+2ℓ into two nontrivial parts, which is X-5-n_2-2n_3+2ℓ≤ X-3-n_2+2ℓ (the latter is guaranteed to be nonnegative for any choice of p). The fixed letters are precisely the letters of B, so there are n_3+1 fixed letters of G_1 and n_3 fixed letters of G_2, with the non-fixed G_1-letters having ≤ (X-n_2-1)/2 degrees of freedom and the non-fixed G_2-letters having ≤ (X-n_2)/2 degrees of freedom. Counting across all choices of values for the letters, p, p', and ℓ,  we have that the number of additional Wicks commutators arising from this case is bounded from above by
 For each choice of p',

    ∑_n_1=0^X-3∑_n_2=0^X-3-n_1∑_ℓ=1^X-2-n_1-n_2/2 (X-3-n_2+2ℓ)  ^X+n_2+1/2   ^X+n_2/2
    ≪   ^X^X,

which is dominated by our error term.

Case 3. Suppose that the fixed points are letters of A. Then, all of the fixed points must be in one of w_1 and w_2; they cannot be in w_3, since then there would be more than 2ℓ letters right of A. First, suppose they are in w_1.  This requires that w_1=D^-1_3^-1AV, where V is the left subword of _1^-1 B _1^-1 DE having length 2ℓ, the length of _2^-1F. The fixed letters are then precisely the letters of A, so there are n_1+1 fixed letters of G_1 and n_1 fixed letters of G_2. The non-fixed G_1-letters have ≤ (X-n_1-1)/2 degrees of freedom and the non-fixed G_2-letters have ≤ (X-n_1)/2  degrees of freedom.The number of possible choices for p' is at most the number of partitions of X-3-m_1=X-3-(2ℓ+2n_1+1+2ℓ)=X-4-2ℓ -n_1 into two nontrivial parts, which is ≤ X-3-2ℓ-n_1 (the latter is guaranteed to be nonnegative for any choice of p).   Counting across all choices of values for the letters, p, p', and ℓ,  we have that the number of additional Wicks commutators arising from this case is bounded from above by

    ∑_n_1=0^X-3∑_n_2=0^X-3-n_1∑_ℓ=1^X-2-n_1-n_2/2 (X-3-2ℓ-n_1)  ^X+n_1+1/2   ^X+n_1/2
    ≪   ^X^X,

which is dominated by our error term.

Finally, suppose the fixed points are in w_2. Since the length difference  2m_1-2ℓ+2 between the lengths of D^-1 and w_1 must be equal to the length difference 2n_1+2ℓ-2m_1-2m_2-2 between the lengths of D^-1_3 A and w_1 '_1 w_2, we require that m_2=n_1+2ℓ-2m_1-2. Note then that p' is parametrized by m_1 ≤ (n_1+2ℓ-2)/2. For a given choice of p', if m_1 ≤ℓ, then we have n_1+1 fixed letters of G_1 and n_1 fixed letters of G_2, with ≤ (X-n_1-1)/2 degrees of freedom for the non-fixed G_1-letters and ≤ (X-n_1)/2 degrees of freedom for the non-fixed G_2-letters. This gives  ≤ (X+n_1+1)/2 ≤ (X+n_1-m_1+ℓ+1)/2 overall degrees of freedom for the G_1-letters, as well as  ≤ (X+n_1-m_1+ℓ)/2 ones for the G_2-letters. If m_1 > ℓ, then we have 2n_1-2(m_1-ℓ)+1 fixed letters of G_1 and 2n_1-2(m_1-ℓ) fixed letters of G_2, so analogously we have ≤ (X+n_1-m_1+ℓ+1)/2 overall degrees of freedom for the G_1-letters, as well as ≤ (X+n_1-m_1+ℓ)/2 ones for for the G_2-letters. Thus, counting across all choices of values for the letters, p, p', and ℓ,  we have that the number of additional Wicks commutators arising from this case is bounded from above by

    ∑_n_1=0^X-3∑_n_2=0^X-3-n_1∑_ℓ=1^X-2-n_1-n_2/2∑_m_1=0^n_1+2ℓ-2/2^X+n_1-m_1+ℓ+1/2   ^X+n_1-m_1+ℓ/2
    ≪   ^X^X,

which is dominated by our error term.
n_1-(ℓ-m_1)  fixed letters if m_1 ≤ℓ

and for each choice of p', we have n_1-(ℓ-m_1) degrees of freedom for the fixed letters and ≤ (X-(n_1-(ℓ-m_1)))/2, for the non-fixed letters.  


Next, we suppose that the fixed letters of g are in two of the three subwords A, B, and E. Consider the following subcases:

Case 1. Suppose the fixed letters of g are in A and B. It is necessary that the fixed letters of A and those of B are in w_i and w_j, respectively, such that i < j; otherwise,  the fixed letters of A would come before the fixed letters of B, a contradiction. First, suppose that the fixed letters of B are in w_2, which implies that the fixed letters of A are in w_1. Then, we require that w_1=D^-1_3AV, where V is the left subword of _1B_1DE having length 2ℓ. This gives 2m_1+1 =2n_1+ 4ℓ +1, i.e, m_1 = n_1+2ℓ. Furthermore, since we have fixed letters of B, we require that V does not include all of B, i.e., 2n_2+2>2ℓ, or equivalently, n_2 ≥ℓ.  Next, for the fixed letters of B to match in position, we require that w_2 ends at the letter b_2n_2-2ℓ+1, which gives us  2m_2+1=(2n_2-2ℓ+1)-(2ℓ+1)+1=2n_2-4ℓ+1, i.e., m_2 = n_2-2ℓ≥ 0. Then, m_3 is automatically determined, and in particular, w_3 is the subword of b_n_2-ℓ+2⋯ b_n_2+1_1 DE omitting the rightmost letter. For this  p' corresponding to p, we have 2n_1+1 fixed letters of A (n_1+1 letters of G_1 and n_1 letters of G_2) and 2n_2-4ℓ+1 fixed letters of B (n_2-2ℓ+1 letters of G_1 and n_2-2ℓ letters of G_2). Next, we bound the degrees of freedom of the non-fixed letters. Note that g maps the letters of EF_3^-1b_1 ⋯ b_ℓ-1b̅_ℓ-1 to those of _3^' -1 b_n_2-ℓ+2⋯ b_n_2+1_1 DE in order, but  g also maps f_ℓ^-1, …, f̅_2^-1 f_1^-1 to b_1 ,…, b̅_ℓ-1, b_ℓ and b_n_2-ℓ+2, …, b_n_2+1, to d_ℓ^-1,…, d_1^-1. Thus, arguing inductively by translation, we see that that choosing the letters of F determines the letters of E, and thus also determines those of D, thereby determining all non-fixed letters (while not caring about the constant number of _i and _i letters). There are ℓ G_1-letters and ℓ-1 G_2-letters in F. Counting across all choices of values for the letters, p, and ℓ, we have that the number of additional Wicks commutators arising from this case is bounded from above by

    ∑_n_1=0^X-3∑_n_2=0^X-3-n_1∑_ℓ=1^X-2-n_1-n_2/2^(n_1+1)+(n_2-2ℓ+1)+ℓ   ^n_1+(n_2-2ℓ)+(ℓ-1)
    ≪   X^X^X,

which is dominated by our error term.

Next, suppose that the fixed letters of B are in w_3. Then, we require that w_3=VB_1 D E, where V is the right subword of D^-1_3 A _1 having length 2n_3-2ℓ+2, the length of EF.  Furthermore, since we have fixed letters of A, we require that V does not include all of A, i.e., 2n_3-2ℓ+2 < 2n_1+2, or in other words, n_3 < n_1+ℓ.  Next, for the fixed letters of A to match in position, we require that they are in w_2, and specifically that w_2=a_n_3-ℓ+1⋯ a_n_1-n_3+ℓ. This gives us 2m_2+1=2n_1-4n_3+4ℓ-1, and thus m_2 =n_1-2n_3+2ℓ-1 > 0. It follows that w_1=D^-1_3 a_1⋯ a_n_3-ℓ. In particular, m_1 is automatically determined, and  for this  p' corresponding to p, we have n_2+1 fixed G_1-letters and n_2 fixed G_2-letters of B, as well as n_1-2n_3+2ℓ fixed G_1-letters and n_1-2n_3+2ℓ-1 fixed G_2-letters of A. Next, we bound the degrees of freedom of the non-fixed letters. Note that g maps the letters of a_n_1-n_3+ℓ+1⋯ a_n_1_2^-1 F^-1 to those of D^-1_3 a_1⋯ a_n_3-ℓ in order. However, g maps d_1, …, d_ℓ, e̅_1, …, e_n_3-2ℓ+1 to a_1, a̅_1, …, a_n_3-ℓ+1, a̅_n_3-ℓ+1. Also, g maps a_n_1-n_3+ℓ+1, ⋯,  a_n_1+1, to e_2, …, e̅_n_3-2ℓ+2, f_1, …, f_ℓ. Thus, arguing inductively by translation, we see that that choosing the letters of F determines the letters of E, and thus also determines those of D, thereby determining all non-fixed letters (while not caring about the constant number of _i and _i letters). Counting across all choices of values for the letters, p, and ℓ, we have that the number of additional Wicks commutators arising from this case is bounded from above by

    ∑_n_1=0^X-3∑_n_2=0^X-3-n_1∑_ℓ=1^X-2-n_1-n_2/2^(n_1-2n_3+2ℓ)+(n_2+1)+ℓ   ^(n_1-2n_3+2ℓ-1)+n_2+(ℓ-1)
    ≪   X^X^X,

which is dominated by our error term.

Case 2. Suppose the fixed letters of g are in B and E. Similarly to before, it is necessary that the fixed letters of B and those of E are in w_i and w_j, respectively, such that i < j.  First, suppose that the fixed letters of B are in w_1. Then, we require that w_1=D^-1_3A_1BV, where V is the left subword of _1DE having length 2n_1+2ℓ+2, the length of _3^-1A_2^-1F^-1.  Then, w_2 must start with e_n_1+3, which means in order to have the letters of E match, we must have 2m_3+3=2n_1+3, i.e., m_3=n_1. Since (2n_1+2)+1+(2m_2+1)+1+(2m_3+1)+1 =2n_3-4ℓ+3 (counting the letters of E in two ways), we have m_2=n_3-2n_1-2ℓ-2>0. There are n_3-2n_1-2ℓ-1 fixed G_1-letters and n_3-2n_1-2ℓ-2 fixed G_2-letters of E in w_2. Also, there are n_2+1 fixed G_1-letters  and n_2 fixed G_2-letters of B. Next, we bound the degrees of freedom of the non-fixed letters. Note that g maps the letters of A_2^-1F^-1 to those of  D e̅_1 ⋯e̅_n_1+1e_n_1+2 in order. 
Likewise, by observing the left end of w_1, we see that g maps the letters of e_n_3-n_1-2ℓ+2⋯e̅_n_3-2ℓ+2 F to those of D^-1_3 A in order. However, we also have that g maps the letters of e̅_1 ⋯e̅_n_1+1e_n_1+2 to those of _3^' -1 e_n_3-n_1-2ℓ+2⋯e̅_n_3-2ℓ+1 e_n_3-2ℓ+2 in order, which overall gives us the ordered equality (by g) of letters

    A_2^-1F^-1e̅_n_3-2ℓ+2F    =D e̅_1 ⋯ e_n_1+2e̅_n_3-2ℓ+2F
    
        = D_3^' -1 e_n_3-n_1-2ℓ+2⋯  e_n_3-2ℓ+2e̅_n_3-2ℓ+2 F=D_3^' -1 D^-1_3A.

Thus, arguing inductively by translation, we see that that choosing the letters of F determines the letters of A, and thus also determines those of D, thereby determining all non-fixed letters (while not caring about the constant number of exceptional letters). Counting across all choices of values for the letters, p, and ℓ, we have that the number of additional Wicks commutators arising from this case is bounded from above by

    ∑_n_1=0^X-3∑_n_2=0^X-3-n_1∑_ℓ=1^X-2-n_1-n_2/2^(n_2+1)+(n_3-2n_1-2ℓ-1)+ℓ   ^n_2+(n_3-2n_1-2ℓ-2)+(ℓ-1)
    ≪   X^X^X,

which is dominated by our error term.


Next, suppose that the fixed  letters of B are in w_2, which implies the fixed letters of E are in w_3. Then, we require that w_3=e_2⋯ e_n_3-2ℓ+2. Furthermore, in order for the letters of B to match in position, we must have that w_2=V_1 B _1 D, where V is the right subword of D^-1_3A having length 2ℓ-1.  We thus have n_3-2ℓ+1 fixed G_1-letters and n_3-2ℓ fixed G_2-letters in E, as well as n_2+1 fixed G_1-letters and n_2 fixed G_2-letters in B. Now, we bound the degrees of freedom of the non-fixed letters. First, suppose that n_1>ℓ. Note then that g maps the letters of F to those of a_n_1-ℓ+2⋯ a_n_1+1 in order, and likewise maps the letters of  A_2^-1F^-1 to those of D _2^' -1 D^-1_3 a_1 ⋯ a_n_1-ℓ+1 in order. Thus, we have the ordered equality  (by g)  of letters 

    a_1 ⋯ a_n_1-ℓ+1a̅_n_1-ℓ+1 F _2^-1F^-1   =  a_1 ⋯ a_n_1-ℓ+1a̅_n_1-ℓ+1 (a_n_1-ℓ+2⋯ a_n_1+1) _2^-1F^-1
       =  D _2^' -1D^-1_3 a_1 ⋯ a_n_1-ℓ-1.

 Thus, arguing inductively by translation, we see that that choosing the letters of F determines the letters of a_1 ⋯ a_n_1-ℓ-1, and thus also determines those of the rest of A and of D, thereby determining all non-fixed letters (while not caring about the constant number of exceptional letters). In the  other case of  n_1 ≤ℓ, the notation above for a_1 ⋯ a_n_1-ℓ-1 becomes inviable, but nevertheless we can use a similar argument as above to conclude that choosing the letters of F determines all the non-fixed letters. Counting across all choices of values for the letters, p, and ℓ, we have that the number of additional Wicks commutators arising from this case is bounded from above by

    ∑_n_1=0^X-3∑_n_2=0^X-3-n_1∑_ℓ=1^X-2-n_1-n_2/2^(n_2+1)+(n_3-2ℓ+1)+ℓ   ^n_2+(n_3-2ℓ)+(ℓ-1)
    ≪   X^X^X,

which is dominated by our error term.

Case 3. Finally, suppose the fixed letters of g are in A and E. Similarly to before, it is necessary that the fixed letters of A and those of E are in w_i and w_j, respectively, such that i < j. First, suppose that the fixed letters of E are in w_3. Then, we require that w_3=e_2⋯ e_n_3-2ℓ+2. Furthermore, in order for the letters of A to match in position, we need that the fixed letters of A are contained in w_1, and in particular, that w_1=D^-1_3A _1 V, where V is the left subword of B_1DE having length 2ℓ-1. We thus have n_3-2ℓ+1 fixed G_1-letters and n_3-2ℓ fixed G_2-letters in E, as well as n_1+1 fixed G_1-letters and n_1 fixed G_2-letters in A.  Now, we bound the degrees of freedom of the non-fixed letters. First, suppose that n_2>ℓ.
 Note then that g maps the letters of F^-1 to those of b_1 ⋯ b_ℓ in order, and likewise maps the letters of F_3^-1 B to those of b_ℓ+1⋯ b_n_2+1_1 D _2^' -1 D^-1. Thus, we have the ordered equality  (by g)  of letters 
 
    F _3^-1 F^-1b̅_ℓ b_ℓ+1⋯ b_n_2+1   = F _3^' -1 (b_1 ⋯ b_ℓ )b̅_ℓ b_ℓ+1⋯ b_n_2+1
       =b_ℓ+1⋯ b_n_2+1_1 D _2^' -1 D^-1

Arguing inductively by translation, we see that that choosing the letters of F determines the letters of B of D, thereby determining all non-fixed letters. In the  other case of  n_2 ≤ℓ, the notation above for b_1 ⋯ b_ℓ becomes inviable, but nevertheless we can use a similar argument as above to conclude that F determines all the non-fixed letters. Counting across all choices of values for the letters, p, and ℓ, we have that the number of additional Wicks commutators arising from this case is bounded from above by

    ∑_n_1=0^X-3∑_n_2=0^X-3-n_1∑_ℓ=1^X-2-n_1-n_2/2^(n_1+1)+(n_3-2ℓ+1)+ℓ   ^n_1+(n_3-2ℓ)+(ℓ-1)
    ≪   X^X^X,

which is dominated by our error term.


Next, suppose that the fixed letters of E are in w_2. Then, the fixed letters of A are contained in w_1, which requires that w_1=D^-1_3A _1 V, where V is the left subword of B_1DE having length 2ℓ-1. But then w_2 must start on a letter not in E, which makes it impossible for the letters of E to match in position. 

Finally, if there are fixed letters in A, B, and C, then it is necessarily that ℓ = 0 and p=p', which does not need to be considered.

If ℓ >(n_3+1)/2, then we can think of our commutator as a cyclic conjugate of C^-1ABCA^-1B^-1 such that the letters are moved from left to right. A symmetric argument like above gives us the conclusion that W'=D^-1_3 A_1 B_1 D E F_2 A^-1_2 B^-1_3 F^-1E^-1 is on average not a generic Wicks commutator of the form w_1 '_1 w_2 '_1 w_3 '_2 w_1^-1'_2 w_2^-1'_3 w_3^-1'_3. Note that this entire argument can then be repeated mutatis mutandis to show that W' is on average also not a Wicks commutator of the other generic form (9) or of the other possible forms (3)-(7), since the only difference from the previous case is a constant number of exceptional letters and possibly setting one or more of the subwords w_1, w_2, and w_3 to be trivial, which overall can only affect error bounds by at worst a multiplicative constant. Likewise, a similar argument shows that W' is also not a Wicks commutator for the case of ℓ=0, again since the only difference from the previous case is a constant number of exceptional letters. Finally, a similar argument shows that even if W is taken to be of the generic form (9) rather than (8), W is on average only decomposable as a Wicks commutator in one prescribed way (i.e., with respect to a unique partition of X-3), and any cyclic conjugate of W is on average not a Wicks commutator. 

We have thus shown that the number of conjugacy classes of commutators with length 4X is given by

    1/6( 1/2(( |G_2|   -2 )^2 +( |G_1|-2 )^2 ) 
       · X^2  ^X-1^X-1 + O( X^X^X))
    
    = 1/12(   ( |G_2|-2 )^2 +( |G_1|-2 )^2 ) 
       · X^2  ^X-1^X-1 +O( X^X^X),

as needed. 



§ CONCLUDING REMARKS


There are a number of directions in which  Theorems <ref> and <ref> can be generalized. First, one can ask: how many conjugacy classes of commutators with word length k are in an arbitrary finitely-generated free product G = G_1 * ⋯ *G_n, with each G_i nontrivial and having symmetric generating set _i={g_1^(i). …, g_m_i^(i),(g_1^(i))^-1. …, (g_m_i^(i))^-1}? While one can define the word length in this context to be with respect to an arbitrary generating set , a natural notion of length to use in this setting would be with respect to the symmetric generating set

    ⋃_i=1^n_i.

We note that in the case that G_i is finite for all 1 ≤ i ≤ n, we can take _i = G_i ∖{1}, for which  is consistent with our choice of the set  of generators for G_1 * G_2 in the statement of Theorem <ref>. Counting conjugacy classes of commutators by word length for groups in this more general  form would have analogous geometric consequences as those discussed in Corollary <ref>. For example,  Hecke Fuchsian groups H() for ≥ 2 have the presentation <cit.>

    ⟨ S,R_ : S^2=I ⟩≅/2 *,

and a desire for geometric corollaries akin to those discussed in Section <ref> motivates a result analogous to Theorem <ref> and  Theorem <ref> in the case of the free product of a nontrivial finite group and an infinite cyclic group. 

To describe a second potential direction for generalization, let the n-commutators of a given group be defined by  the elements with trivial abelianization and commutator length n. A second direction for generalizing Theorems <ref> and <ref> is to, for any n, count the number of conjugacy classes of n-commutators with word length k in a group G taken to be either the free group F_r or a finitely generated free product. This is natural to ask, given that Culler <cit.> has classified the possible forms of n-commutators for a free group and Vdovina <cit.> has done this for an arbitrary free product. In fact, Culler has also classified the possible forms that a product of n square elements can take for a free group, so an analogous question can be asked for the number of conjugacy classes comprised of products of n square elements. If one obtained the asymptotic number of n-commutators with length k (respectively, of n-square-element-products with length k), this would also give a corollary analogous to Corollary <ref>. Specifically, given a connected CW-complex X with fundamental group G, one would obtain the asymptotic number of free homotopy classes of loops γ : S^1 → X with length k (in the generators of ) such that there exists a genus-n orientable surface Y with one boundary component (respectively,  a connected sum  Y of n real projective planes, with one boundary component) and a continuous map f : Y → X satisfying f(∂ Y) = γ, but also that this statement does not hold when replacing n with any m ≤ n. We expect our combinatorial method to also work in this generalized setting, as long as one has, for the given group, an explicit list of the possible Wicks forms of n-commutators (or of products of n square elements).

One important question regarding the growth of groups is whether or not a given growth series S(z) ∑_X=0^∞ a_X z^X is a rational function. For S(z) to be rational, it is necessary that 

    a_X ∼∑_i=1^n c_i X^m_iα_i^X

for some complex numbers α_1,…, α_n,c_1,…,c_n and nonnegative integers m_1,…, m_n, as a function of X <cit.>. Consequently, estimates on the number of conjugacy classes of commutator-subgroup elements in F_r of length k show that its growth series cannot be rational <cit.>. On the other hand, we have shown that the number of conjugacy classes of commutators of length 2X in F_r (or those of length 4X in G_1*G_2) has an asymptotic that is of the desired form (<ref>), which raises the question of whether or not its growth series is rational.



§ ACKNOWLEDGMENTS


 This work began at Princeton University as part of the author's senior thesis advised by Peter Sarnak, whom the author would like to thank for providing the number-theoretic motivation for this problem, invaluable discussions/references, and constant encouragement. 

The generalization of Theorem <ref> to an arbitrary free product of two nontrivial finite groups was done at Harvard University and was supported by the National Science Foundation Graduate Research Fellowship Program (grant number DGE1745303). The author would like to thank Bena Tshishiku for reading over the manuscript and providing many valuable suggestions. He would also like to thank Aaron Calderon, Keith Conrad, Noam Elkies, Curt McMullen, Hector Pasten, Xiaoheng Wang, and Boyu Zhang for  very helpful discussions.




§ ACKNOWLEDGMENTS


This work began at Princeton University as part of the author's senior thesis advised by Peter Sarnak, whom the author would like to thank for providing the number-theoretic motivation for this problem, invaluable discussions/references, and constant encouragement. 

The generalization of Theorem <ref> to an arbitrary free product of two nontrivial finite groups was done at Harvard University and was supported by the National Science Foundation Graduate Research Fellowship Program (grant number DGE1745303). The author would like to thank Bena Tshishiku for reading over the manuscript and providing many valuable suggestions. He would also like to thank Aaron Calderon, Keith Conrad, Noam Elkies, Curt McMullen, Hector Pasten, Xiaoheng Wang, and Boyu Zhang for  very helpful discussions.

amsplain


