Density of systoles of hyperbolic manifolds
Sami Douba Junzhi Huang








We show that for each n ≥ 2, the systoles of closed hyperbolic n-manifolds form a dense subset of (0, +∞). We also show that for any n≥ 2 and any Salem number λ, there is a closed arithmetic hyperbolic n-manifold of systole log(λ). In particular, the Salem conjecture holds if and only if the systoles of closed arithmetic hyperbolic manifolds in some (any) dimension fail to be dense in (0, +∞).




Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France
douba@ihes.fr

Department of Mathematics, Yale University, New Haven, CT 06511, USA
junzhi.huang@yale.edu

Density of systoles of hyperbolic manifolds
    Junzhi Huang
    April 29, 2024
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The systole sys(M) of a closed nonpositively curved Riemannian manifold M is the length of a shortest closed geodesic in M. 
It follows for instance from residual finiteness of surface groups that there is a sequence of closed hyperbolic surfaces with systole going to infinity. Since for any ϵ > 0 and any closed surface M of negative Euler characteristic there is a hyperbolic metric on M of systole < ϵ, continuity of the systole function on moduli space then yields for any L>0 a closed hyperbolic surface of systole precisely L.





By Mostow rigidity <cit.>, the set of systoles of closed hyperbolic n-manifolds is countable for n ≥ 3. Nevertheless, we show the following. 




  For any n≥ 2, the set of systoles of closed hyperbolic n-manifolds is dense in (0,+∞).


That 0 is an accumulation point of systoles of closed hyperbolic 3-manifolds follows for instance from Thurston's hyperbolic Dehn filling theory (see <cit.>). The analogous fact for 4-manifolds was established by Agol <cit.>, whose strategy was to “inbreed” arithmetic manifolds along totally geodesic hypersurfaces. This strategy was successfully extended to arbitrary dimensions independently by Belolipetsky–Thomson <cit.> and Bergeron–Haglund–Wise <cit.>. The present note is an elaboration on the utility of this inbreeding technique.

We also establish a variant of Theorem <ref> for arithmetic hyperbolic manifolds. A Salem number is a real algebraic integer λ > 1 of degree ≥ 4 that is Galois-conjugate to λ^-1 and all of whose remaining Galois conjugates lie on the unit circle. 


Let λ > 1 be a Salem number. Then for any n ≥ 2, there is a closed arithmetic hyperbolic n-manifold of systole precisely log(λ).


The manifolds we exhibit in the proof of Theorem <ref> are  arithmetic of simplest type (the latter being the only arithmetic construction that applies in every dimension) and are moreover classical in the language of Emery–Ratcliffe–Tschantz <cit.>. For n ≥ 4, it follows from Meyer's theorem on quadratic forms and <cit.> that the exponential lengths of closed geodesics in any classical simplest-type closed arithmetic hyperbolic n-manifold are Salem numbers.[Exponential geodesic lengths in closed arithmetic hyperbolic surfaces or 3-manifolds can be quadratic “Salem numbers”; the latter are sometimes included in the definition of a Salem number. Exponential geodesic lengths in odd-dimensional closed arithmetic hyperbolic orbifolds need not be Salem numbers even in this more general sense; see Lemma 4.10 and Theorem 4.11 in <cit.>, as well as <cit.>.]

Lehmer's conjecture for Salem numbers, or the Salem conjecture for short, asserts that the Salem numbers are bounded away from 1; see <cit.> and <cit.>. In light of the relationship between Salem numbers and lengths of geodesics in arithmetic hyperbolic manifolds, Theorem <ref> yields the following reformulation of the Salem conjecture.


The Salem conjecture holds if and only if, for some, equivalently, any, n ≥ 2, the set of systoles of closed arithmetic hyperbolic n-manifolds fails to be dense in (0, +∞). 

 It is known generally that, given any irreducible symmetric space X of noncompact type, the Salem conjecture implies a uniform lower bound on systoles of closed arithmetic locally symmetric manifolds modeled on X; see Fra̧czyk–Pham <cit.>. On the other hand, since positive powers of Salem numbers remain Salem numbers, failure of the Salem conjecture would imply density of Salem numbers in (1, +∞), and hence density of systoles of closed arithmetic hyperbolic n-manifolds in (0, +∞) for each n ≥ 2 by Theorem <ref>.





We motivate the proofs of Theorems <ref> and <ref> with the following remark. 



We provide an argument along the lines of <cit.> that, given any L > 0, there is a closed hyperbolic surface of systole precisely L. Indeed, it suffices to find a closed hyperbolic surface M_L of systole ≥ L possessing two disjoint 2-sided simple closed geodesics Σ_1 and Σ_2 and an orthogeodesic segment ω joining the Σ_i of length L/2 and whose length is minimal among all orthogeodesic segments with endpoints on Σ_1 ∪Σ_2. Cutting M_L along the Σ_i and then doubling the resulting surface along its boundary yields a (possibly disconnected) closed hyperbolic surface of systole precisely L. 

One can construct such a surface M_L as follows. Let P ⊂ℍ^2 be a right-angled pentagon with an edge ω of length precisely L/2, and let H_1, H_2 ⊂ℍ^2 be the walls of P adjacent to ω. Let Γ_P < Isom(ℍ^2) be the group generated by the reflections in the walls of P. We can find a larger right-angled convex polygon Q ⊂ℍ^2 that is a union of finitely many Γ_P-translates of P such that the walls of Q that enter the L/2-neighborhood of H_i are orthogonal to H_i for i =1,2 (this idea originates in work of Scott <cit.>; see also <cit.>). By residual finiteness and virtual torsion-freeness of the group Γ_Q < Isom(ℍ^2) generated by the reflections in the walls of Q, there is a finite-index subgroup Λ <Γ_Q such that Λ\ℍ^2 is a surface of systole ≥ L. We may now take M_L = Λ\ℍ^2, the Σ_i to be the projections to M_L of the H_i, and ω to be the projection of ω.








Given L, ϵ>0, we  exhibit a closed hyperbolic n-manifold M with |sys(M)-L|<ϵ. As in Remark <ref>, it suffices to find a closed hyperbolic n-manifold M_L, ϵ of systole ≥ L + ϵ possessing two disjoint 2-sided properly embedded totally geodesic hypersurfaces Σ_1 and Σ_2, and an orthogeodesic segment ω joining the Σ_i of length within ϵ/2 from L/2 and whose length is minimal among all orthogeodesic segments with endpoints on Σ_1 ∪Σ_2. We can then cut M_L, ϵ along Σ_1∪Σ_2, select the component N containing ω of the resulting manifold with boundary, and define M to be the double of N. The double of ω is indeed a shortest closed geodesic in M of length within ϵ from  L.

To that end, let k=ℚ(√(2)) and _k = ℤ[√(2)]. Define f to be the quadratic form on ^n+1 given by

    f(x_1,⋯,x_n+1)=x_1^2+⋯+x_n^2-√(2)x_n+1^2.


We identify the level set {x∈^n+1| f(x)=-1, x_n+1>0} with n-dimensional hyperbolic space ^n and '(f;) with Isom(^n), where '(f;) is the index-2 subgroup of (f;) preserving ^n. By the Borel–Harish-Chandra theorem <cit.>, we have that Γ:= '(f;_k) is a uniform arithmetic lattice of Isom(^n). 

Denote by H_1 the hyperplane {x_1=0} in ^n. Since '(f;k) is dense in '(f;), there is an element g∈'(f;k) such that

    |dist_^n(H_1,gH_1)-L/2| < ϵ/2.

Let ω be the orthogeodesic segment in ^n connecting H_1 and H_2:= gH_1. By the proof of Lemma 3.1 in Belolipetsky–Thomson  <cit.>, we may now pass to a non-zero ideal I ⊂𝒪_k such that dist_^n(H_1, γ H_2) ≥dist_^n(H_1, H_2) for each γ in the principal congruence subgroup Γ(I)<Γ of level I.
Up to diminishing the ideal I, we can further assume that


  * the principal congruence subgroup Γ(I) is torsion-free, so that M_L, ϵ:= Γ(I) \^n is a manifold;

  *  we have sys(M_L, ϵ) ≥ L + ϵ;

  *  for i=1,2 and γ∈Γ(I), either γ H_i = H_i, in which case γ furthermore preserves each side of H_i, or dist_^n(H_i, γ H_i) ≥dist_^n(H_1, H_2); in particular, the projections Σ_i of the H_i to M_L, ϵ are 2-sided and properly embedded.




Note that item (<ref>) can be arranged by residual finiteness of the ring 𝒪_k, whereas item (<ref>) can be ensured by the fact that the stabilizer in Γ of either side of H_i is an intersection of congruence subgroups of Γ for i=1,2; see <cit.> and <cit.>. We now have that the Σ_i are disjoint and that any orthogeodesic segment in M_L, ϵ with endpoints on Σ_1 ∪Σ_2 has length at least that of the projection ω of ω to M_L,ϵ. Thus, the manifold M_L, ϵ, the hypersurfaces Σ_i, and the orthogeodesic segment ω are as desired.





The proof of Theorem <ref> demonstrates that we may take the closed hyperbolic manifolds giving rise to a dense set of systoles in (0, +∞) to be quasi-arithmetic in the sense of Vinberg and all share the same adjoint trace field (indeed, the same Vinberg ambient algebraic group); see <cit.> and compare <cit.>. On the other hand, by varying the form f in the proof, one easily produces a family of pairwise incommensurable closed hyperbolic manifolds in each dimension whose systoles remain dense in (0, +∞). 

In fact, it is possible to achieve the latter without varying f; one can select from any family of closed hyperbolic manifolds whose systoles are dense in (0, +∞) pairwise incommensurable manifolds whose systoles remain dense in (0, +∞). Indeed, given a closed hyperbolic manifold M, the set of all geodesic lengths of manifolds commensurable to M is closed and discrete in [0, +∞). When M is nonarithmetic, the latter follows from Margulis's arithmeticity criterion <cit.>, and when M is arithmetic, from the fact that, for fixed D >0 and μ > 1, there are only finitely many monic integer polynomials of degree ≤ D and Mahler measure ≤μ; see the discussion immediately following Conjecture 10.2 in <cit.>. 




Instead of appealing to the work of Belolipetsky–Thomson (loc. cit.) to find the manifold M_L, ϵ in the proof of Theorem <ref>, we could have instead used <cit.> as in the proof of Theorem 4 in <cit.>. An interesting feature of the former approach is that it afforded us a manifold M_L, ϵ that is congruence arithmetic.







Let μ = λ + λ^-1. Let k= ℚ(μ), and 𝒪_k be the ring of integers of k. Following the proof of <cit.>, we define the form f in n+1 variables over k to be that given by the symmetric matrix

    [     1         μ/2;       I_n-1      ;   μ/2           1 ].

The form f is of signature (n,1), so that we may again identify ^n with one of the two sheets of the hyperboloid {x∈^n+1| f(x)=-1} and Isom(^n) with '(f;), where '(f;) is the index-2 subgroup of (f;) preserving ^n. Moreover, the field k is totally real, and for any embedding σ:k→ other than the identity (of which at least one exists), the form f^σ is positive definite. It follows that Γ := '(f ; 𝒪_k) is a uniform arithmetic lattice of '(f ; ℝ)= Isom(^n), which contains the reflections

    τ_1 := [    -1          -μ;       I_n-1      ;                 1 ], τ_2 := [    -μ       1-μ^2;       I_n-1      ;     1           μ ]

whose respective fixed hyperplanes H_1, H_2 ⊂^n satisfy 
    dist_^n(H_1, H_2) = log(λ)/2.
  As in the proof of Theorem <ref>, by the aforementioned work of Belolipetsky–Thomson (loc. cit.), we may pass to a non-zero ideal I ⊂𝒪_k such that


  * the principal congruence subgroup Γ(I) is torsion-free, so that M_log(λ):= Γ(I) \^n is a manifold;

  * we have sys(M_log(λ)) ≥log(λ);

  * the H_i project to disjoint properly embedded 2-sided totally geodesic hypersurfaces Σ_1 and Σ_2 in M_log(λ); 

  * any orthogeodesic segment in M_log(λ) with endpoints on Σ_1 ∪Σ_2 has length ≥log(λ)/2. 

Now cutting M_log(λ) along the Σ_i and then doubling the resulting manifold along its boundary yields a (possibly disconnected) closed manifold M of systole precisely log(λ) each of whose components is arithmetic (since τ_i ∈Γ for i=1,2). 







 §.§ Acknowledgements
 We thank Misha Belolipetsky, Bram Petri, and Alan Reid for helpful comments. S.D. was supported by the Huawei Young Talents Program. This work was initiated during a visit of S.D. to Yale University in March 2024; S.D. thanks Sebastian Hurtado and the entire geometry/topology/dynamics team at Yale for their hospitality.





siam

