Cone types and asymptotic invariants for the random walk on the modular group
Angel Pardo




  
  
  





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        On the random walk on PSL(2,𝐙)]
        Cone types and asymptotic invariants for the random walk on the modular groupThis work was supported by ANID-Chile through the FONDECYT 1221934 grant.
        Departamento de Matemática y Ciencia de la Computación,
        Universidad de Santiago de Chile,
        Las Sophoras 173, Estación Central,
        Santiago, Chile.
        angel.pardo@usach.cl
        
        
        We compute the cone types of the Cayley graph of the modular group  associated with the standard system of generators {[  0 -1;  1  0 ],[ 1 1; 0 1 ]}.
        We do this by showing that, in general, there is a set of suffixes of each element that completely determines the cone type of the element, and such suffixes are subwords of primitive relators.
        
        Then, using J. W. Cannon's seminal ideas (1984), we compute its growth function.
        We estimate from above and below the spectral radius of the random walk using ideas from T. Nagnibeda (1999) and S. Gouëzel (2015).
        Finally, using results of Y. Guivarc'h (1980) and S. Gouëzel, F. Mathéus and F. Maucourant (2015), we estimate other asymptotic invariants of the random walk, namely, the entropy and the drift.
        
        [
            Angel Pardo
            December 30, 2023
        =====================
        
        
        
        § INTRODUCTION
        
        
        The modular group  is arguably one of the most fundamental groups in mathematics and, accordingly, one of the most studied.
        
        In this note we study  through the lens of combinatorial and geometric group theory, focusing on the asymptotic invariants for the (simple symmetric) random walk on the Cayley graph of  associated with the standard generators r=[  0 -1;  1  0 ] and u=[ 1 1; 0 1 ]. 
        
        
        It is worth to mention that  is also the free product of the cyclic groups generated by r and ru=[  0 -1;  1  1 ], of order two and three, respectively.
        
        Thus, we have the presentation ≅_2 * _3 ≅⟨ a,b | a^2,b^3⟩.
        From the combinatorial and geometric group theoretic view point, this presentation is the handier.
        In fact, the invariants for  associated with the generators r and ru can be computed explicitly, as we survey in the <Ref>.
        
        
        We focus our attention in the generators r and u as they do not allow such straightforward computations but are still of interest. In fact, they are geometrically meaningful as orbifold fundamental curves on the modular curve \ (as it is also the case for the generating set {ru,u}, which we address in <Ref>).
        Our case of study is also of interest from the combinatorial and geometric group theoretic point of view, since it is still a Cayley graph of a free product of two cyclic groups, but with respect to a generating set different from the ones that have been well understood.
        
        We base our method on J. W. Cannon's classification of group elements by their cone types and our main contribution is the computation of the cone types of  relative to {r,u}.
        This allows to compute the growth series using Cannon's original ideas <cit.>, and to give numerical estimates for the spectral radius of the random walk using ideas from T. Nagnibeda <cit.> and S. Gouëzel <cit.>.
        We also estimate the entropy and the drift of the random walk applying a result of S. Gouëzel, F. Mathéus and F. Maucourant <cit.>.
        
        
        One motivation for writing this note is that, even if it is likely that in the present case estimates (or even a complete description; cf. <Ref>) may be known by experts, we could not find any clue in the literature.
        On the other hand, this work arises as a spin-off of <cit.>, where we compute cone types for a different Fuchsian group in order to apply Nagnibeda's ideas and give lower bounds for the bottom of the spectrum of the combinatorial Laplace operator, which is equivalent to estimates from above for the spectral radius of the corresponding random walk, as shows <ref> below.
        
        In particular, there is some text overlap in respect to some background material.
        We also stress that the estimates in the present work do not intend to be optimal in any sense.
        
        
        
         §.§ Cone types
        
        For a set A, we denote by A^* the free monoid on A, an element in A^* is called a word in A, and the elements in A are the letters.
        
        For a system of generators S of a group G, in a slight abuse of notation we still write S^* for (S ∪ S^-1)^*.
        There is a natural evaluation morphism S^* → G, that associates to a word w ∈ S^* the product of its letters in G.
        We denote the length of an element of S^* by |·|, that is, the number of its letters.
        Similarly, we denote by |·|_S the word length in G associated with S.
        That is, for an element g ∈ G, |g|_S = min{|w| w ∈^-1(g)}.
        We say that w ∈ S^* is a geodesic if |w| = |(w)|_S.
        We also denote inversion by an overbar, that is, if g ∈ G, then g̅ = g^-1∈ G and, if w = w_1 … w_n ∈ S^*, where w_1,…,w_n ∈ S, then w̅ = w_n^-1… w_1^-1∈ S^*.
        
        
        Given g ∈ G, the cone of g relative to S, denoted (g), is the set of h ∈ G for which some geodesic from 𝕀 to gh passes through g, that is,
        
            (g) = {h ∈ G∃ v,w ∈ S^*, (v) = g, (w) = h, |vw| = |gh|_S}.
        
        For each element g ∈ G, its cone (g) naturally defines a rooted subgraph of the Cayley graph.
        We say that two cones _1, _2 are equivalent (or, of the same type) if they are isomorphic as rooted graphs.
        
        The cone type of g ∈ G (relative to S) is the equivalence class of (g), which, in a slight abuse of notation, we still denote by (g).
        
        It follows from Cannon's work that if G is a word-hyperbolic group —as is the case of the modular group —, then there are only finitely many cone types <cit.>. 
        Our proof follows the same lines but does not depend on this result, however.
        
        
        Let G= and S = {r,u}.
        Then, there are exactly six cone types:
        (𝕀), (r), (u), (ru), (ur) and (rur).
        
        
        
        Furthermore, we give a simple combinatorial description for each cone type in <Ref>.
        On the other hand, in the proof of <Ref> arises the following general phenomenon of independent interest.
        
        
        Let G be a group generated by a finite system S ⊂ G and suppose that the set of primitive relators (over S) is finite.
        Then, G has finitely many cone types (relative to S).
        
        
        Here, by relator we mean a word in S that evaluates to the identity and by primitive, that it does not contain proper subwords that are also relators.
        
        In fact, we show in <Ref>, that for any group G and generating system S ⊂ G, the set of all maximal suffixes of geodesics for g ∈ G that are subwords of a primitive or trivial relator completely determines the cone type of g.
        In particular, knowing the set of primitive relators allows to compute, algorithmically at least, every cone type.
        
        
        
         §.§ Growth
        
        For n ∈, let _n ⊂ G be the sphere of radius n, that is, _n = {g ∈ G |g|_S = n}.
        The (spherical) growth series of G relative to S is then the formal series
        
            V(x) = ∑_g ∈ G x^|g| = ∑_n = 0^∞ |_n|  x^n.
        
        It follows from Cannon's work that if G has finitely many cone types relative to S, then the growth series corresponds to a rational analytic function <cit.> and the reciprocal of its radius of convergence gives the rate of exponential growth of the group G relative to S, denoted v(G,S) and also called growth, for short.
        
        
        Let G= and S = {r,u}.
        Then, the growth series of G relative to S corresponds to the rational analytic function
        
            (1+x)(1+x+x^2)/1-x-x^2.
        
        In particular, the rate of exponential growth of G relative to S is the golden ratio, that is,
        
            v(G,S) = φ = 1 + √(5)/2≈
            1.61 803.
        
        In particular, the growth sequence (|_n|)_n ∈ corresponds to <cit.>.
        
        
         §.§ Random walk
        
        The (right, simple symmetric) random walk on G relative to S is the Markov chain on G whose transition probabilities are defined by
        
            p(g,h) = 1/|S∪S̅|    if g̅h ∈ S ∪S̅, 
             0     otherwise.
        
        A realization of the random walk starting from the identity is given by X_0 = 𝕀∈ G and X_n = s_1… s_n, where (s_i)_i is an independent sequence of (S ∪S̅)-valued uniformly distributed random variables.
        In other words, it is the (simple symmetric) random walk on the Cayley graph of G relative to S.
        
        
        
        
          §.§.§ Spectral radius
        
        We denote _Sℓ^2(G)→ℓ^2(G), the corresponding Markov operator, that is,
        
            (_S h)_g 1/|S ∪S̅|∑_s∈ S ∪S̅ h_gs,
                
            h∈ℓ^2(G).
        
        We denote ρ(G,S) the spectral radius of the random walk on G relative to S, that is, the spectral radius of _S,
        
            ρ(G,S) = sup{|⟨_S h, h ⟩/⟨ h, h ⟩|,  h∈ℓ^2(G)}.
        
        We estimate ρ(G,S) from above following ideas of Nagnibeda <cit.>.
        We also follow ideas of Gouëzel <cit.> to estimate ρ(G,S) from below.
        More precisely, we prove the following.
        
        
        Let G= and S = {r,u}.
        Then, the spectral radius of the random walk on G relative to S satisfies
        
            0.976 336
            < ρ(G,S) <
            
            0.976 642
            .
        
        The spectral radius associated with a symmetric finite system of d>1 generators, is bounded from below by the spectral radius of (the random walk on) a regular tree of degree d, that is, by 2√(d-1)/d (see, e.g., <cit.>).
        In our case, this yields
        ρ(G,S) ≥ 2√(2)/3 ≈
        0.942 809.
        In particular, our estimates from below improves the trivial bound obtained by comparison to the regular tree.
        
        It is also worth to mention that the spectral radius of G relative to S' = {r,ru} can be explicitly computed.
        In fact, 
        ρ(G,S') = 1/6 + 1/6√(13 + 8 √(2))≈
        0.988 482
        (see <Ref>).
        
        
          §.§.§ Bottom of the spectrum of the Laplace operator
        
        A related quantity of interest is the bottom of the Laplace spectrum, which we denote μ_0(G,S).
        
        More precisely, let Δ_Sℓ^2(G)→ℓ^2(G) be the Laplace operator on G relative to S, that is,
        
            (Δ_S h)_g ∑_s∈ S∪S̅ (h_g - h_gs),
                
            h∈ℓ^2(G).
        
        Then μ_0(G,S) is the bottom of the spectrum of Δ_S, that is,
        
            μ_0(G,S) = inf{⟨Δ_S h, h ⟩/⟨ h, h ⟩,  h∈ℓ^2(G)}.
        
        
        By definition, Δ_S = |S ∪S̅|(_ℓ^2(G) - _S).
        Moreover, when the Cayley graph is bipartite —as is the case for G= and S = {r,u}—, the spectrum of the Markov operator is symmetric.
        In such case, it follows that the bottom of the spectrum μ_0(G,S) is related to the spectral radius ρ(G,S) by the formula 
        μ_0(G,S) = |S ∪ S^-1| (1-ρ(G,S)).
        
        
        As a direct consequence of <ref> and <Ref> we get the following.
        
        Let G= and S = {r,u}.
        Then, the bottom of the spectrum of the Laplace operator on G relative to S satisfies
        
            0.0 700 754
            < μ_0(G,S) <
            
            0.0 709 903
            .
        
        Compare with the trivial upper bound given by the regular tree of degree three, that is,
        μ_0(G,S) ≤ 3 - 2√(2)≈
        0. 171 572.
        Also, compare with the bottom of the Laplace spectrum relative to S' = {r, ru}: 
        μ_0(G,S') = 5/2 - 1/2√(13 + 8 √(2))≈
        0.0 345 534
        (see <Ref>).
        
        
          §.§.§ Entropy and drift
        
        Several numerical quantities have been introduced to describe the asymptotic behavior of random walks on groups (relative to a generating system). The spectral radius ρ(G,S) is one of them.
        Other important asymptotic invariants are the (asymptotic) entropyh(G,S) and the driftℓ(G,S) of the random walk. These are defined by
        
            h(G,S) = lim_n→∞ -1/n∑_g ∈ Gυ_S^*n(g) log(υ_S^*n(g))
                and    ℓ(G,S) = lim_n→∞1/n∑_g ∈ G |g| υ_S^*n(g),
        
        where υ_S is the uniform distribution on S ∪S̅ and υ_S^*n, its n-fold convolution.
        There is a fundamental inequality relating the entropy, the drift and the growth due to Guivarc'h <cit.>.
        Moreover, Gouëzel, Mathéus and Maucourant <cit.> showed that estimates on the asymptotic invariants can be obtained from the others in several ways.
        This, together with <Ref>, allows us to prove the following.
        
        
        Let G= and S = {r,u}.
        Then, the entropy and the drift of the random walk on G relative to S satisfy
        
            0.0 938 046
            < h(G,S) <
            
            0.347 676
                and    
            0.0 579 744
            < ℓ(G,S) <
            
            0.214 876
            .
        
        
         §.§ Comparison between the three generating systems
        
        In <Ref> we study other two geometrically meaningful systems of generators of the modular group .
        Namely, S' = {r,t} and S” = {t,u}, where r = [  0 -1;  1  0 ], t = [  0 -1;  1  1 ] and u = [ 1 1; 0 1 ].
        Note that t = ru.
        For a schematic comparison, we summarize in <Ref> the results on the asymptotic invariants associated with these three generating systems.
        
        
        
         §.§ Structure of the paper
        
        In  <Ref>, we recall the basics of combinatorial group theory we need and prove some useful general combinatorial results. In particular, in <Ref> we prove <Ref>.
        
        In <Ref>, we compute the cone types of G = relative to S = {r,u}, proving <Ref>, and use them to produce a recurrence for the growth series as in Cannon's seminal work.
        We prove <Ref> in <Ref> and <Ref>, in <Ref>.
        We also include a graphical representation of the Cayley graph associated with S in <Ref>.
        
        In <Ref> we estimate the spectral radius ρ(G,S).
        In <Ref>, we state Nagnibeda's ideas for the upper bounds and, in <Ref>, those of Gouëzel, for the lower bounds.
        To conclude the proof of <Ref>, we include some details of the computations for the lower bound in <Ref>.
        
        In <Ref> we state Guivarc'h <cit.> and Gouëzel–Mathéus–Maucourant <cit.> results relating the entropy and the drift to the growth and the spectral radius. Using our estimates for the spectral radius ρ(G,S) and the exact value of the growth v(G,S), this allows us to estimate the entropy h(G,S) and the drift ℓ(G,S) and prove <Ref>.
        
        In <Ref> we survey the analogous results in the case of the system of generators S' = {r,t=ru}, which yields a presentation of  as a free product of cyclic groups. In such case much more can be proven. In particular, we present the full Markov spectrum for the simple symmetric random walk in  relative to S' and the exact values of the corresponding growth, spectral radius, entropy and drift.
        We also include a graphical representation of the Cayley graph associated with S' in <Ref>.
        
        In <Ref> we repeat the study done for S = {r,u}, for the system of generators S” = {t=ru,u}.
        That is, we compute the growth series, estimate the spectral radius, the entropy and the drift of  relative to S”.
        We also include a graphical representation of the Cayley graph associated with S” in <Ref>.
        
        
        
         §.§ Acknowledgements
        
        The author is grateful Sebastien Gouëzel for the reference to the work of T. Nagnibeda.
        The author is also greatly indebted to Tatiana Nagnibeda for useful comments and suggestions that improved the presentation of the paper, and for her patience providing him insight on the problems discussed in this work.
        
        
        
        § COMBINATORIAL GROUP THEORY
        
        In this section, we recall the basics of combinatorial group theory we need and prove some useful general combinatorial results useful for our purposes.
        
        
        The following discussion is completely general.
        For a complete introduction to this topic we refer the reader to Magnus–Karrass–Solitar's book <cit.>.
        
        Let G be any group, and let S be a subset of G. A word in S is any expression of the form
        
            w=s_1^σ _1s_2^σ _2… s_n^σ _n
        
        where s_1,…,s_n∈ S and σ_i ∈{+1,-1}, i=1,…,n. The number |w|=n is the length of the word.
        The empy word, denoted , is the only one word of length zero.
        
        Given two words v and w, we say that v is a subword of w if w=xvy, for some words x and y. If x is the empty word we say that v is a prefix of w. If y is the empty word we say that v is a suffix of w.
        
        Each word w in S represents an element of G.
        Namely, the product of the expression, which we denote (w) ∈ G.
        For example, the identity element can be represented by the empty word, that is, 𝕀 = ().
        We say that two words are equivalent if they represent the same element in G.
        
        
        As usual, we use exponential notation for abbreviation; for example, the word sss can also be denoted by s^3.
        We also use an overbar to denote inverses, thus s̅ stands for s^-1, 
        and if w = s_1^σ _1… s_n^σ _n, then w̅ = s_n^-σ _n… s_1^-σ _1.
        
        
        In these terms, a subset S of a group G is a system of generators if and only if every element of G can be represented by a word in S.
        Henceforth, let S be a fixed system of generators of G and a word is assumed to be a word in S.
        We say that a non-empty word is a relator if it represents the identity element of G, that is, if it equivalent to the empty word.
        A generator next to its own inverse (ss̅ or s̅s) define a trivial relator.
        
        For a relator, we call a subword that is also a relator, a subrelator.
        We say that a word is reduced if it has no trivial subrelators.
        We say that a non-trivial relator is primitive if if it does not contain proper subrelators.
        In particular, a word is a geodesic if and only if it contains no primitive relators as subword.
        Note that, if S is a system of generators and P is the set of all primitive relators on S, then ⟨ S | P ⟩ is a presentation of G.
        
        For an element g∈ G, we consider the word norm|g|_S to be the least length of a word which represents g when considered as a product in G, and every such word is called a geodesic, that is, if its length coincides with its word norm when considered as a product in G.
        A geodesic does not contains relators as subwords and, in particular, it is is always reduced.
        Also, a subword of a geodesic is also a geodesic.
        
        
        The following decomposition result (see <Ref>) will be useful in <Ref> (in the form of <Ref> below).
        
        
        Let v,w be two different equivalent geodesics. Then, there are geodesics v_0,v_1, w_0,w_1 and x such that v = v_0v_1x and w = w_0w_1x, and v_1w̅_1 is a primitive relator (of even length).
        
        Let x be the largest common suffix of v and w (possibly x is empty). Write v=v'x and w = w'x.
        Let w_1 and v_1 be the smallest non-empty suffixes of w' and v' respectively such that v_1 and w_1 are equivalent.
        Such v_1 and w_1 exist since v and w are different words.
        Moreover, they have the same length since they are equivalent geodesics, that is, they evaluate to the same element in G.
        Write v'=v_0v_1 and w'=w_0w_1 (possibly v_0 and w_0 are empty).
        In particular v_0 and w_0 are equivalent, since the same holds for v',w' and v_1,w_1.
        See <Ref>.
        
        It remains to prove that v_1w̅_1 is primitive.
        Suppose z is a subrelator of v_1w̅_1.
        Since v_1 and w_1 are geodesics, 
        
        their subwords are also geodesics and, therefore, they do not contain any relator as subword.
        It follows that z = v_2w̅_2 for some non-empty suffixes v_2 and w_2, of v_1 and w_1, respectively. In particular, v_2 and w_2 are non-empty suffixes of w' and v', respectively, and v_2,w_2 are equivalent. But, by definition, v_1 and w_1 are the smallest such suffixes and therefore v_2=v_1 and w_2=w_1. Thus, v_1w̅_1 has no proper subrelators and, therefore, v_1w̅_1 is primitive.
        
        
        As a direct consequence of <Ref>, we have the following.
        
        
        Let v=v'yx and w=w'zx be two equivalent geodesics such that y z̅ is also geodesic. Then, yz̅ is a subword of some primitive relator (of even length).
        
        
        Consider the decomposition given by <Ref>. It is clear that y is a subword of v_1 and z, of w_1. Then yz̅ is a subword of the primitive relator v_1w̅_1.
        
        
        
        
        
         §.§ Cone types
        
        Recall that the cone of an element g ∈ G, relative to S, is the set (g) of h ∈ G for which some geodesic from 𝕀 to gh passes through g.
        More precisely,  let (g) be the set of all geodesics for g ∈ G, that is, (g) = {w ∈ S^*π(w) = g, |w| = |g|_S}.
        Then,
        
            (g) = {h ∈ G∃ w ∈(g), v ∈(h)  wv ∈(gh)}.
        
        Moreover, in a slight abuse of notation, we shall identify (g) with the rooted graph that it defines as subgraph of the Cayley graph of G.
        Finally, the cone type of an element g ∈ G is the isomorphism class of rooted graphs of (g).
        
        To any rooted graph Γ, one can associate its tree of geodesics, that is, the (rooted) tree where vertices are geodesics from the root of Γ and edges correspond to length-increasing edges in Γ.
        It is clear that isomorphic rooted graphs yield isomorphic trees of geodesics.
        
        In particular, in order to study cone types it is worth to determine their tree of geodesics.
        A way to do this is through other type functions.
        A type function for G (relative to S) is a function τ G → T, for some set T of types, such that the type of an element τ(g) determines the number of successors of each type.
        In other words, two elements g, g' ∈ G have the same type if and only if, for every t∈ T,
        
            |{s∈ S_+(g) τ(gs)=t}| = |{s'∈ S_+(g') τ(g's')=t}|
            ,
        
        where S_±(g) = {s ∈ S ∪S̅ |gs| = |g| ± 1}.
        
        Thus, the tree of geodesics of cone types are, in a precise way, the minimal type functions.
        In fact, it is straightforward from the definition that, for every g ∈ G, the type τ(g) completely determines the tree of geodesics of (g).
        
        Consequently, once we have determined a type function, in order to completely determine the cone type of an element, one has to be able to detect geodesics representing the same vertex and possible length-preserving edges.
        In the case where every relator has even length, there are no length-preserving edges and equivalent geodesics can be characterized using primitive relators through <Ref>, for example.
        
        
        In the presence of relators of odd length, it is convenient to be able to determine the type of neighbors of the same length as well.
        Thus, an upgraded definition of type function in such case would include in addition the condition that, for every g,g' ∈ G sharing the same type and for every type t ∈ T, |{s∈ S_0(g) τ(gs)=t}| = |{s'∈ S_0(g') τ(g's')=t}|, where S_0(g) = {s ∈ S ∪S̅ |gs| = |g|}.
        
        
        
        The previous discussion motivates the following.
        
        Let ϕ S^* → S^* be such that ϕ(w) is the longest suffix of w ∈ S^* that is a subword of some primitive or trivial relator (we need to consider trivial relators in the presence of free letters).
        
        
        Let g,h ∈ G.
        Then, h ∈(g) if and only if there is v ∈(h) such that ϕ(w)v is a geodesic for every w ∈(g).
        
        Suppose that h ∈(g), that is, there is w_0 ∈(g) and v ∈(h) such that w_0v ∈(gh). In particular, |gh|_S = |g|_S + |h|_S.
        Now, for every w ∈(g), |wv| = |w| + |v| = |g|_S + |h|_S = |gh|_S and therefore wv ∈(gh).
        Thus, ϕ(w)v is also a geodesic as subword of the geodesic wv.
        
        Suppose now that h ∉(g), that is, wv is not a geodesic for every w ∈(g) and every v ∈(h).
        
        It follows that, for every w ∈(g) and every v ∈(h), there is a minimal non-empty suffix w' of w and a minimal non-empty prefix v' of v such that w'v' is subword of a relator.
        Now, choose w'v' also minimal among all w ∈(g) and v ∈(h).
        Then, similarly to the proof of <Ref>, we can show that w'v' is in fact subword of either a primitive or trivial relator.
        It follows that w' is a suffix of ϕ(w) and, therefore, ϕ(w)v is not a geodesic.
        
        
        Let (g) = ϕ((g)), that is, (g) is the set of all maximal suffixes of geodesics for g that are subwords of a primitive or trivial relator.
        Then, by <Ref>, we have the following.
        
        
        The map G → 2^S^* determines the cone types.
        
        If there is a finite number of primitive relators, then  takes values in a finite set.
        By <Ref>, it follows that there is a finite number of cone types.
        
        
        Note that <Ref> not only shows that  determines the cone type of an element, but the cone itself.
        Moreover, by construction, it is possible to determine the -type of the successors of a given -type.
        In fact, we have the following.
        
        
        Let g ∈ G. Then s ∈ S_+(g) if and only if s ∈ S_+((w)) for every w ∈(g).
        Moreover, for s ∈ S_+(g), (gs) = ((ϕ(ws))) for any w ∈(g).
        
        
        § CONE TYPES FOR THE MODULAR GROUP
        
        The previous discussion is completely general.
        We now specialize to the modular group G= with generators S = {r,u}, where r=[  0 -1;  1  0 ] and u=[ 1 1; 0 1 ].
        
        In order to compute the cone types, we start by computing , which is a type function by <Ref>.
        
        It is well known that ⟨ r,u | r^2,(ru)^3⟩ is a presentation of G (see, e.g., <cit.>).
        Since we have the relator r^2, 
        we can omit henceforth r̅, as it coincides with r as element in G.
        The set of primitive relators is then 
        given by
        
            { r^2, (r u)^3, (r u̅)^3, (ur)^3, (u̅r)^3}.
        
        
        Denote (g) the set of suffixes of geodesics for g ∈ G.
        Then, by the description of the primitive relators, as a direct consequence of <Ref>, we have the following.
        
        
        The following cases cannot happen:
        
            ∙      u,u̅∈(g),
               ∙      ur,u̅r ∈(g),
            ∙      ar,a̅^2 ∈(g),
               ∙      ar,a ∈(g),
               for  a     = u  or u̅.
        
        Neither u^2, uru̅ nor u̅ru are subwords of a primitive relator.
        
        
        It follows that  can only take values in the set
        
            {∅,{r},{a},{ra},{ar},{ra,a},{rar,a̅ra̅}}_a=u,u̅
        
        and any such value is possible.
        
        
        The -type of the successors of a given -type is determined as in <Ref>.
