Extending the analogous result of Cannon and Wagreich for the fundamental groups of surfaces, we show that, for the l-regular graphs X associated to regular tessellations of hyperbolic plane by m-gons, the denominators of the growth series (which are rational and were computed by Floyd and Plotnick) are reciprocal Salem polynomials. As a consequence, the growth rates of these graphs are Salem numbers. We then derive some regularity properties for the coefficients $a_n$ of the growth series: they satisfy $$Kλ^n-R<a_n<Kλ^n+R$$ for some constants $K,R>0$, $λ>1$.
In this paper we prove that there is a direct relationship between Salem numbers and translation lengths of hyperbolic elements of arithmetic hyperbolic groups that are determined by a quadratic form over a totally real number field. As an application we determine a sharp lower bound for the length of a closed geodesic in a noncompact arithmetic hyperbolic n-orbifold for each dimension n. We also discuss a "short geodesic conjecture", and prove its equivalence with "Lehmer's conjecture" for Salem numbers.
It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show that any non-compact arithmetic $3$-dimensional orbifold defines $c Q^{1/2} + O(Q^{1/4})$ square-rootable Salem numbers of degree $4$ which are less than or equal to $Q$. This quantity can be compared to the total number of such Salem numbers, which is shown to be asymptotic to $\frac{4}{3}Q^{3/2}+O(Q)$. Assuming the gap conjecture of Marklof, we can extend these results to compact arithmetic $3$-orbifolds. As an applic…
Counting Salem numbers arising from arithmetic hyperbolic orbifolds
University of Minnesota（明尼苏达大学）；Universidade Federal Fluminense（弗鲁米嫩塞联邦大学）；Centro de Investigación en Matemáticas AC（数学研究中心AC）；Universiteit Utrecht（乌得勒支大学）
Let G be a non-elementary torsion-free hyperbolic group. We prove that the exponential growth rate of the periodic quotient G/G^n tends to the one of G as n odd approaches infinity. Moreover we provide an estimate at which the convergence is taking place.
We show that every non-elementary group $G$ acting properly and cocompactly by isometries on a proper geodesic Gromov hyperbolic space $X$ is growth tight. In other words, the exponential growth rate of $G$ for the geometric (pseudo)-distance induced by $X$ is greater than the exponential growth rate of any of its quotients by an infinite normal subgroup. This result generalizes from a unified framework previous works of Arzhantseva-Lysenok and Sambusetti, and provides an answer to a question of the latter.
Rivin conjectured that the conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. Ciobanu, Hermiller, Holt and Rees proved that the conjugacy growth series of a virtually cyclic group is rational. Here we present the proof confirming the other direction of the conjecture, by showing that the conjugacy growth series of a non-elementary hyperbolic group is transcendental. We also present and prove some variations of Rivin's conjecture for commensurability classes and primitive conjugacy classes. We then explore Rivin's conjecture for finitely gene…
Generalising results of Razborov and Safin, and answering a question of Button, we prove that for every hyperbolic group there exists a constant $α>0$ such that for every finite subset $U$ that is not contained in a virtually cyclic subgroup $|U^n|\geqslant (α|U|)^{[(n+1)/2]}$. Similar estimates are established for groups acting acylindrically on trees or hyperbolic spaces.
We prove that every non-elementary hyperbolic group $G$ acts with maximal growth on some set $X$ such that every orbit of any element $g \in G$ is finite. As a side-product of our approach we prove that if $G$ is non-elementary hyperbolic, $\HH \leq G$ is quasiconvex of infinite index then there exists $g \in G$ such that $<\HH,g>$ is quasiconvex of infinite index and is isomorphic to $\HH*<g >$ if and only if $\HH \cap E(G)= \{e\} $, where $E(G)$ is the maximal finite normal subgroup of $G$.
We introduce a new invariant of finitely generated groups, the ambiguity function, and prove that every finitely generated acylindrically hyperbolic group has a linearly bounded ambiguity function. We use this result to prove that the relative exponential growth rate $\lim \limits_{n \rightarrow \infty} \sqrt[n]{\vert B^{X}_H(n) \vert}$ of a subgroup $H$ of an acylindrically hyperbolic group $G$ exists with respect to every finite generating set $X$ of $G$, if $H$ contains a loxodromic element of $G$. Further we prove that the relative exponential growth rate of every finitely generated subgro…
The following short note provides an alternative proof of a result of Coornaert: namely, that given a non-elementary word-hyperbolic group $G$ with a finite generating set $X$, there exist constants $λ,D > 1$ such that \[ D^{-1}λ^n \leq |B_{G,X}(n)| \leq D λ^n \] for all $n \geq 0$, where $B_{G,X}(n)$ is the ball of radius $n$ in the Cayley graph $Γ(G,X)$.
Consider the tesselation of the hyperbolic plane by m-gons, l per vertex. In its 1-skeleton, we compute the growth series of vertices, geodesics, tuples of geodesics with common extremities. We also introduce and enumerate "holly trees", a family of reduced loops in these graphs. We then apply Grigorchuk's result relating cogrowth and random walks to obtain lower estimates on the spectral radius of the Markov operator associated with a symmetric random walk on these graphs.
Let $H$ be a hyperbolic group, $A$ and $B$ be subgroups of $H$, and $gr(H,A,B)$ be the growth function of the double cosets $AhB, h \in H$. We prove that the behavior of $gr(H,A,B)$ splits into two different cases. If $A$ and $B$ are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as $gr(H,A,B)$. We can even take $A=B$. In contrast, for quasiconvex subgroups A and B of infinite index, $gr(H,A,B)$ is exponential. Moreover, there exists a constant $λ> 0$, such that $gr(H,A,B)(r) >λf_H(r)$ for all big enough $r$, where $f_H(r)$ is the growth function…
We show that the set $SA(G)$ of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic group $G$ is dense in the product of the sets $SA(P)$ over all maximal parabolic subgroups $P$. The set $BSA(G)$ of equivalence classes of biautomatic structures on $G$ is isomorphic to the product of the sets $BSA(P)$ over the cusps (conjugacy classes of maximal parabolic subgroups) of $G$. Each maximal parabolic $P$ is a virtually abelian group, so $SA(P)$ and $BSA(P)$ were computed in ``Equivalent automatic structures and their boundaries'' by M.Shapiro and W.Neuman…
Floyd showed that if a sequence of compact hyperbolic Coxeter polygons converges, then so does the sequence of the growth rates of the Coxeter groups associated with the polygons. For the case of the hyperbolic 3-space, Kolpakov discovered the same phenomena for specific convergent sequences of hyperbolic Coxeter polyhedra. In this paper, we show that the growth rate is a continuous function on the space of Coxeter systems. This is an extension of the results due to Floyd and Kolpakov since the convergent sequences of Coxeter polyhedra give rise to that of Coxeter systems in the space of marke…
We give several sufficient conditions for uniform exponential growth in the setting of virtually torsion-free hierarchically hyperbolic groups. For example, any hierarchically hyperbolic group that is also acylindrically hyperbolic has uniform exponential growth. In addition, we provide a quasi-isometric characterization of hierarchically hyperbolic groups without uniform exponential growth. To achieve this, we gain new insights on the structure of certain classes of hierarchically hyperbolic groups. Our methods give a new unified proof of uniform exponential growth for several examples of gro…
We recast the notion of joint spectral radius in the setting of groups acting by isometries on non-positively curved spaces and give geometric versions of results of Berger-Wang and Bochi valid for $δ$-hyperbolic spaces and for symmetric spaces of non-compact type. This method produces nice hyperbolic elements in many classical geometric settings. Applications to uniform growth are given, in particular a new proof and a generalization of a theorem of Besson-Courtois-Gallot.
This paper proves that in a non-elementary relatively hyperbolic group, the logarithm growth rate of any non-elementary subgroup has a linear lower bound by the logarithm of the size of the corresponding generating set. As a consequence, any non-elementary subgroup has uniform exponential growth.
We introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the classes of groups known to have locally uniform exponential growth to include extensions of either word hyperbolic or right-angled Artin groups by groups with locally uniform exponential growth. From this, we deduce that the automorphism group of a torsion-free one-ended hyperbolic group has locally uniform exponential growth. Our methods also demonstrate that…
It is shown that the growth rate $(\lim_r |B(r)|^{1/r})$ of any $k$ faces Dirichlet tiling of the real hyperbolic space $\mathbb{H}^d, d>2,$ is at most $k-1-ε$, for an $ε> 0$, depending only on $k$ and $d$. We don't know if there is a universal $ε_u > 0$, such that $k-1-ε_u$ upperbounds the growth rate for any $k$-regular tiling, when $ d > 2$?
We show that if $H$ is a quasiconvex subgroup of infinite index in a non-elementary hyperbolic group $G$ then the Schreier coset graph $X$ for $G$ relative to $H$ is non-amenable (that is, $X$ has positive Cheeger constant). We present some corollaries regading the Martin boundary and Martin compactification of $X$ and the co-growth of $H$ in $G$.
We give a quantification of residual finiteness for the fundamental groups of hyperbolic manifolds that admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Specifically, we give explicit upper bounds on residual finiteness that are linear in terms of geodesic length. We then extend the linear upper bounds to hyperbolic manifolds with a finite cover that admits such an immersion. Since the quantifications are given in terms of geodesic length, we define the geodesic residual finiteness growth and show that this growth is equivalent to the usual re…
Given a periodic quotient of a torsion-free hyperbolic group, we provide a fine lower estimate of the growth function of any sub-semi-group. This generalizes results of Razborov and Safin for free groups.
This note addresses some questions that arise in the series of works by Kyoji Saito on the growth functions of graphs. We study "hyperbolike" graphs, which include Cayley graphs of hyperbolic groups. We generalize some well-known results on hyperbolic groups to the hyperbolike setting, including rationality of generating functions, and sharp estimates on the growth rate of vertices. We then apply these results to confirm a conjecture of Saito on the "opposite series", which was originally posed for hyperbolic groups.
We prove that for a relatively hyperbolic group G there is a sequence of relatively hyperbolic proper quotients such that their growth rates converge to the growth rate of G. Under natural assumptions, the same conclusion holds for the critical exponent of a cusp-uniform action of G on a hyperbolic metric space. As a corollary, we obtain that the critical exponent of a torsion-free geometrically finite Kleinian group can be arbitrarily approximated by those of quotient groups. This resolves a question of Dal'bo-Peigné-Picaud-Sambusetti. Our approach is based on the study of Patterson-Sullivan …
We prove that the mapping torus group $\FN \rtimes_α \Z$ of any automorphism $α$ of a free group $\FN$ of finite rank $n \geq 2$ is weakly hyperbolic relative to the canonical (up to conjugation) family $\mathcal H(α)$ of subgroups of $\FN$ which consists of (and contains representatives of all) conjugacy classes that grow polynomially under iteration of $α$. Furthermore, we show that $\FN \rtimes_α \Z$ is strongly hyperbolic relative to the mapping torus of the family $\mathcal H(α)$. As an application, we use a result of Drutu-Sapir to deduce that $\FN \rtimes_α \Z$ has Rapic Decay.
We prove a general version of the amenability conjecture in the unified setting of a Gromov hyperbolic group G acting properly cocompactly either on its Cayley graph, or on a CAT(-1)-space. Namely, for any subgroup H of G, we show that H is co-amenable in G if and only if their exponential growth rates (with respect to the prescribed action) coincide. For this, we prove a quantified, representation-theoretical version of Stadlbauer's amenability criterion for group extensions of a topologically transitive subshift of finite type, in terms of the spectral radii of the classical Ruelle transfer …
We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, non-elementary group $G$ acting on a $G$--space $\mathcal{X}$, we prove that if $G$ contains a strongly contracting element and if $G$ is not too badly distorted in $\mathcal{X}$, then the action of $G$ on $\mathcal{X}$ is a growth tight action. It follows that if $\mathcal{X}$ is a cocompact, relatively hyperbolic $G$--space, then the action of $G$ on $\mathcal{X}$ is a growth tight action. This g…
We prove that the semistability growth of hyperbolic groups is linear, which implies that hyperbolic groups which are sci (simply connected at infinity) have linear sci growth. Based on the linearity of the end-depth of finitely presented groups we show that the linear sci is preserved under amalgamated products over finitely generated one-ended groups. Eventually one proves that most non-uniform lattices have linear sci.
We introduce an obstruction to the existence of a coarse embedding of a given group or space into a hyperbolic group, or more generally into a hyperbolic graph of bounded degree. The condition we consider is "admitting exponentially many fat bigons", and it is preserved by a coarse embedding between graphs with bounded degree. Groups with exponential growth and linear divergence (such as direct products of two groups one of which has exponential growth, solvable groups that are not virtually nilpotent, and uniform higher-rank lattices) have this property and hyperbolic graphs do not, so the fo…
We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron-Frobenius theory. We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions use the notion of growth tightness and the work of Dahmani, Guirardel, and Osin on rotating families.
We give an effective upper bound, for certain arithmetic hyperbolic 3-manifold groups obtained from a quadratic form construction, on the minimal index of a subgroup that embeds in a fixed 6-dimensional right-angled reflection group, stabilizing a totally geodesic subspace. In particular, for manifold groups in any fixed commensurability class we show that the index of such a subgroup is asymptotically smaller than any fractional power of the volume of the manifold. We also give effective bounds on the geodesic residual finiteness growths of closed hyperbolic manifolds that totally geodesicall…
We define a large class of abstract Coxeter groups, that we call $\infty$--spanned, and for which the word growth rate and the geodesic growth rate appear to be Perron numbers. This class contains a fair amount of Coxeter groups acting on hyperbolic spaces, thus corroborating a conjecture by Kellerhals and Perren. We also show that for this class the geodesic growth rate strictly dominates the word growth rate.
