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.
The conjugacy growth function counts the number of distinct conjugacy classes in a ball of radius $n$. We give a lower bound for the conjugacy growth of certain branch groups, among them the Grigorchuk group. This bound is a function of intermediate growth. We further proof that certain branch groups have the property that every element can be expressed as a product of uniformly boundedly many conjugates of the generators. We call this property bounded conjugacy width. We also show how bounded conjugacy width relates to other algebraic properties of groups and apply these results to study the …
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.
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 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.
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 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…
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…
We show that every non-decreasing function $f\colon \mathbb N\to \mathbb N$ bounded from above by $a^n$ for some $a\ge 1$ can be realized (up to a natural equivalence) as the conjugacy growth function of a finitely generated group. We also construct a finitely generated group $G$ and a subgroup $H\le G$ of index 2 such that $H$ has only 2 conjugacy classes while the conjugacy growth of $G$ is exponential. In particular, conjugacy growth is not a quasi-isometry invariant.
In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is $0$ for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups, and the lamplighter group.
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 cla…
Given a group $G$ and a family of subgroups $\mathcal{F}$, we consider its classifying space $E_{\mathcal F}G$ with respect to $\mathcal{F}$. When $\mathcal F = \mathcal{VC}yc$ is the family of virtually cyclic subgroups, Juan-Pineda and Leary conjectured that a group admits a finite model for this classifying space if and only if it is virtually cyclic. By establishing a connection to conjugacy growth we can show that this conjecture holds for linear groups. We investigate a similar question that was asked by Lück--Reich--Rognes--Varisco for the family of cyclic subgroups. Finally, we constru…
Modelled on efficient algorithms for solving the conjugacy problem in hyperbolic groups, we define and study the permutation conjugacy length function. This function estimates the length of a short conjugator between words $u$ and $v$, up to taking cyclic permutations. This function might be bounded by a constant, even in the case when the standard conjugacy length function is unbounded. We give applications to the complexity of the conjugacy problem, estimating conjugacy growth rates, and languages. Our main result states that for a relatively hyperbolic group, the permutation conjugacy lengt…
Let $G$ be a finite group. We consider the problem of counting simultaneous conjugacy classes of $n$-tuples and simultaneous conjugacy classes of commuting $n$-tuples in $G$. Let $α_{G,n}$ denote the number of simultaneous conjugacy classes of $n$-tuples, and $β_{G,n}$ the number of simultaneous conjugacy classes of commuting $n$-tuples in $G$. The generating functions $A_G(t) = \sum_{n\geq 0} α_{G,n}t^n,$ and $B_G(t) = \sum_{n\geq 0} β_{G,n}t^n$ are rational functions of $t$. We show that $A_G(t)$ determines and is completely determined by the class equation of $G$. We show that $α_{G,n}$ gro…
In this note we produce examples of outer automorphisms of finitely generated groups which have exotic behaviors in terms of growth of conjugacy classes.
We study generalisations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for $\mathcal{C}$-conjugacy separability of a wreath product $A \wr B$ in terms of the $\mathcal{C}$-conjugacy separability of $A$ and $B$, the growth of $\mathcal{C}$-cyclic subgroup separability of $B$, and the $\mathcal{C}$-residual girth of $B.$ As an application, we provide a characterisation of when $A \wr B$ is $p$-conjugacy separable. We use this characterisation to the provide for each prime $p$ an example of wreath products with infinite base group that are …
We show that any subgroup of a finitely generated virtually abelian group $G$ grows rationally relative to $G$, that the set of right cosets of any subgroup of $G$ grows rationally, and that the set of conjugacy classes of $G$ grows rationally. These results hold regardless of the choice of finite weighted generating set for $G$.
We develop new computational methods for studying potential counterexamples to the Andrews-Curtis conjecture, in particular, Akbulut-Kurby examples AK(n). We devise a number of algorithms in an attempt to disprove the most interesting counterexample AK(3). To improve metric properties of the search space (which is a set of balanced presentations of the trivial group) we introduce a new transformation (called an ACM-move here) that generalizes the original Andrews-Curtis transformations and discuss details of a practical implementation. To reduce growth of the search space we introduce a strong…
In this paper, we derive an asymptotic formula for the number of conjugacy classes of elements in a class of statistically convex-cocompact actions with contracting elements. Denote by $\mathcal C(o, n)$ (resp. $\mathcal C'(o, n)$) the set of (resp. primitive) conjugacy classes of pointed length at most $n$ for a basepoint $o$. The main result is an asymptotic formula as follows: $$\sharp \mathcal C(o, n) \asymp \sharp \mathcal C'(o, n) \asymp \frac{\exp(ω(G)n)}{n}.$$ A similar formula holds for conjugacy classes using stable length. As a consequence of the formulae, the conjugacy growth serie…
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 $H$.
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…
For a Coxeter system $(W,S)$ let $a_n^{(W,S)}$ be the cardinality of the sphere of radius $n$ in the Cayley graph of $W$ with respect to the standard generating set $S$. It is shown that, if $(W,S)\preceq(W',S')$ then $a_n^{(W,S)}\leq a_n^{(W',S')}$ for all $n\in \mathbb{N}_0$, where $\preceq$ is a suitable partial order on Coxeter systems (cf. Thm. A). It is proven that there exists a constant $τ= 1.13\dots$ such that for any non-affine, non-spherical Coxeter system $(W,S)$ the growth rate $ω(W,S)=\limsup \sqrt[n]{a_n}$ satisfies $ω(W,S)\geq τ$ (cf. Thm. B). The constant $τ$ is a Perron numbe…
