Graph products of cyclic groups and Coxeter groups are two families of groups that are defined by labeled graphs. The family of Dyer groups contains these both families and gives us a framework to study these groups in a unified way. This paper focuses on the growth series of a Dyer group $D$ with respect to the standard generating set. We give a recursive formula for the growth series of $D$ in terms of the growth series of standard parabolic subgroups. As an application we obtain the rationality of the growth series of a Dyer group. Furthermore, we show that the growth series of $D$ is closely related to the Euler characteristic of $D$.
The principal objects studied in this note are Coxeter groups $W$ that are neither finite nor affine. A well known result of de la Harpe asserts that such groups have exponential growth. We consider quotients of $W$ by its parabolic subgroups and by a certain class of reflection subgroups. We show that these quotients have exponential growth as well. To achieve this, we use a theorem of Dyer to construct a reflection subgroup of $W$ that is isomorphic to the universal Coxeter group on three generators. The results are all proved under the restriction that the Coxeter diagram of $W$ is simply l…
In this paper we introduce the geodesic conjugacy language and geodesic conjugacy growth series for a finitely generated group. We study the effects of various group constructions on rationality of both the geodesic conjugacy growth series and spherical conjugacy growth series, as well as on regularity of the geodesic conjugacy language and spherical conjugacy language. In particular, we show that regularity of the geodesic conjugacy language is preserved by the graph product construction, and rationality of the geodesic conjugacy growth series is preserved by both direct and free products.
In this paper we consider groups of the form $G\wr L$, where the set of generators naturally extends the sets of generators of $G$ and $L$, and $L$ admits a Cayley graph that is a tree. We show how one can compute the conjugacy growth series of such groups in terms of the standard and conjugacy growth series of $G$. We then provide explicit formulas for groups of the form $G\wr \mathbb{Z}$ and $G\wr (C_2*C_2)$. We also prove that the radius of convergence of the conjugacy growth series of $G\wr L$, for any $G$ and $L$ as above, is the same as the radius of convergence of its standard growth se…
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 $K,R>0$, $λ>1$.
It is observed that the conjugacy growth series of the infinite fini-tary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed, for other generating sets, for restricted permutational wreath products of finite groups by the finitary symmetric group, and for alternating groups. Similar methods are used to compute usual growth polynomials and conjugacy growth polynomials for finite symmetric groups and alternating groups, with respect to various generating sets of transpositions. Comput…
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.
We study the HNN extension of $\mathbb{Z}^m$ given by the cubing endomorphism $g\mapsto g^3$, and prove that such groups have rational growth. To do so, we describe a method of computing the subgroup growth series of the horocyclic subgroup $\mathbb{Z}^m$ in this family of examples, prove that for all $m$ the group has rational growth. In the appendix, the subgroup growth series has been computed for all $m \leq 10$.
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…
In this paper we give asymptotics for the conjugacy growth of the soluble Baumslag-Solitar groups $BS(1,k)$, $k\geq 2$, with respect to the standard generating set, by providing a complete description of geodesic conjugacy representatives. We show that the conjugacy growth series for these groups are transcendental, and give formulas for the series. As a result of our computation we also establish that in each $BS(1,k)$ the conjugacy and standard growth rates are equal.
We introduce an algorithmic framework to investigate spherical and geodesic growth series of braid groups relatively to the Artin's or Birman-Ko-Lee's generators. We present our experimentations in the case of three and four strands and conjecture rational expressions for the spherical growth series with respect to the Birman-Ko-Lee's generators.
Several classical formulae for the growth series of a Coxeter group are proved in a new way, using the structure of the Coxeter complex, the Davis complex, or the Tits non-complex.
In the first, mostly expository, part of this paper, a graded Lie algebra is associated to every group G given with an N-series of subgroups. The asymptotics of the Poincare series of this algebra give estimates on the growth of the group G. This establishes the existence of a gap between polynomial growth and growth of type $e^{\sqrt n}$ in the class of residually-p groups, and gives examples of finitely generated p-groups of uniformly exponential growth. In the second part, we produce two examples of groups of finite width and describe their Lie algebras, introducing a notion of Cayley graph…
We show that the geodesic growth function of any finitely generated virtually abelian group is either polynomial or exponential; and that the geodesic growth series is holonomic, and rational in the polynomial growth case. In addition, we show that the language of geodesics is blind multicounter.
In this note we employ infinite ergodic theory to derive estimates for the algebraic growth rate of the Poincaré series for a Kleinian group at its critical exponent of convergence.
A group is said to hae a rational growth with respect to the generating set if the growth series is a rational polynomial. It was shown by Parry that a subset of torus bundle groups exhibits rational growth. We generalize this result to other torus bundle groups.
We calculate asymptotic estimates for the conjugacy growth function of finitely generated class 2 nilpotent groups whose derived subgroup is infinite cyclic, including the so-called higher Heisenberg groups. We prove that these asymptotics are stable when passing to commensurable groups, by understanding their twisted conjugacy growth. We also use these estimates to prove that, in certain cases, the conjugacy growth series cannot be a holonomic function.
I describe a class of groups acting on rooted trees. The original claim was that all have intermediate word growth between polynomial and exponential. The argument constructs a functional equation on the growth formal power series, and derives the growth properties from its analytical behaviour. Unfortunately, there are substantial mistakes in the proofs, so the main claims should be considered unsettled. In particular, it is still an open problem whether the growth of the Gupta-Sidki group is intermediate or exponential.
We provide upper and lower bounds on the length of the shortest non-trivial element in the derived series and lower central series in the free group on two generators. The techniques are used to provide new estimates on the nilpotent residual finiteness growth and on almost laws for compact groups.
We examine the conjugacy growth series of all wreath products of the finitary permutation groups $\text{Sym}(X)$ and $\text{Alt}(X)$ for an infinite set $X$. We determine their asymptotics, and we characterize the limiting behavior between the $\text{Alt}(X)$ and $\text{Sym}(X)$ wreath products. In particular, their ratios form a limit if and only if the dimension of the symmetric wreath product is twice the dimension of the alternating wreath product.
In this paper we give a recursive formula for the conjugacy growth series of a graph product in terms of the conjugacy growth and standard growth series of subgraph products. We also show that the conjugacy and standard growth rates in a graph product are equal provided that this property holds for each vertex group. All results are obtained for the standard generating set consisting of the union of generating sets of the vertex groups.
This paper explores the nature of the solution sets of systems of equations in virtually abelian groups. We view this question from two angles. From a formal language perspective, we prove that the set of solutions to a system of equations forms an EDT0L language, with respect to a natural normal form. Looking at growth, we show that the growth series of the language of solutions is rational. Furthermore, considering the set of solutions as a set of tuples of group elements, we show that it has rational relative growth series with respect to any finite generating set.
Let $G$ be a finitely generated group with a finite generating set $S$. For $g\in G$, let $l_S(g)$ be the length of the shortest word over $S$ representing $g$. The growth series of $G$ with respect to $S$ is the series $A(t) = \sum_{n=0}^\infty a_n t^n$, where $a_n$ is the number of elements of $G$ with $l_S(g)=n$. If $A(t)$ can be expressed as a rational function of $t$, then $G$ is said to have a rational growth function. We calculate explicitly the rational growth functions of $(p,q)$-torus link groups for any $p, q > 1.$ As an application, we show that their growth rates are Perron number…
We give explicit formulas for the geodesic growth series of a Right Angled Coxeter Group (RACG) based on a link-regular graph that is 4-clique free, i.e. without tetrahedrons
The objective of this paper is to detect which combinatorial properties of a regular graph can completely determine the geodesic growth of the right-angled Coxeter or Artin group this graph defines, and to provide the first examples of right-angled and even Coxeter groups with the same geodesic growth series.
We develop further Cannon's method of cone types for finding the growth function of a group, which can also be used to find the coordination sequences of certain infinite graphs. We then apply this method to compute the growth functions and series of the wallpaper groups (the 2 dimensional crystallographic groups). The paper has a number of illustrating colored figures and tables summarizing the results.
In this paper we derive a generating series for the number of cellular complexes known as pavings or three-dimensional maps, on $n$ darts, thus solving an analogue of Tutte's problem in dimension three. The generating series we derive also counts free subgroups of index $n$ in $Δ^+ = \mathbb{Z}_2*\mathbb{Z}_2*\mathbb{Z}_2$ via a simple bijection between pavings and finite index subgroups which can be deduced from the action of $Δ^+$ on the cosets of a given subgroup. We then show that this generating series is non-holonomic. Furthermore, we provide and study the generating series for isomorphi…
In this paper we study the smallest Mealy automaton of intermediate growth, first considered by the last two authors. We describe the automatic transformation monoid it defines, give a formula for the generating series for its (ball volume) growth function, and give sharp asymptotics for its growth function, namely [ F(n) \sim 2^{5/2} 3^{3/4} π^{-2} n^{1/4} \exp{π\sqrt{n/6}} ] with the ratios of left- to right-hand side tending to 1 as $n \to \infty$.
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.
This paper, which is the last of a series of three papers, studies dynamical properties of elements of $\mathrm{Out}(F_{\tt n})$, the outer automorphism group of a nonabelian free group $F_{\tt n}$. We prove that, for every subgroup $H$ of $\mathrm{Out}(F_{\tt n})$, there exists an element $ϕ\in H$ such that, for every element $g$ of $F_{\tt n}$, the conjugacy class $[g]$ has polynomial growth under iteration of $ϕ$ if and only if $[g]$ has polynomial growth under iteration of every element of $H$.
Recently we have shown a structure theorem for locally compact groups of polynomial growth. We give now some applications on various growth functions and relations to FC-G - series. In addition, we show some results on related classes of groups.
In this paper, we consider the formal power series whose n-th coefficient is the number of copies of a given finite graph in the ball of radius n centred at the identity element in the Cayley graph of a finitely generated group and call it the growth function. Epstein, Iano-Fletcher and Uri Zwick proved that the growth function is a rational function if the group has a geodesic automatic structure. We compute the growth function in the case where the group is abelian and see that the denominator of the rational function is determined from the rank of the group.
We study norm convergence and summability of Fourier series in the setting of reduced twisted group $C^*$-algebras of discrete groups. For amenable groups, Følner nets give the key to Fejér summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable gr…
We show that the characteristic series for the greedy normal form of a Coxeter group is always a rational series, and prove a reciprocity formula for this series when the group is right-angled and the nerve is Eulerian. As corollaries we obtain many of the known rationality and reciprocity results for the growth series of Coxeter groups as well as some new ones.
