Related papers: Conjugacy growth series and languages in groups
To every finitely generated group one can assign the conjugacy growth function that counts the number of conjugacy classes intersecting a ball of radius $n$. Results of Ivanov and Osin show that the conjugacy growth function may be constant…
We initiate the study of the \emph{twisted conjugacy growth series} of a finitely generated group, the formal power series associated to the twisted conjugacy growth function. Our main result is that, for a virtually abelian group, this…
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…
We describe an effective version of the conjugacy problem and study it for wreath products and free solvable groups. The problem involves estimating the length of short conjugators between two elements of the group, a notion which leads to…
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…
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…
Given a finitely generated group with generating set $S$, we study the cogrowth sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to one. This is related to the probability of return for walks the…
A group presentation is said to have rational growth if the generating series associated to its growth function represents a rational function. A long-standing open question asks whether the Heisenberg group has rational growth for all…
In this paper we generalise and unify the results and methods used by Benson, Liardet, Evetts, and Evetts & Levine, to show that rational sets in a virtually abelian group G have rational (relative) growth series with respect to any…
A regular set of words is ($k$-)locally testable if membership of a word in the set is determined by the nature of its subwords of some bounded length $k$. In this article we study groups for which the set of all geodesic words with respect…
Let $\Gamma$ be the fundamental group of a manifold modeled on three dimensional Sol geometry. We prove that $\Gamma$ has a finite index subgroup $G$ which has a rational growth series with respect to a natural generating set. We do this by…
We study local piecewise conjugacy of the quantized dynamics arising from factorial languages. We show that it induces a bijection between allowable words of same length and thus it preserves entropy. In the case of sofic factorial…
The density of a rational language can be understood as the frequency of some "pattern" in the shift space, for example a pattern like "words with an even number of a given letter." We study the density of group languages, i.e. rational…
This note records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely…
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…
The complexity of a geodesic language has connections to algebraic properties of the group. Gilman, Hermiller, Holt, and Rees show that a finitely generated group is virtually free if and only if its geodesic language is locally excluding…
Determining the length of short conjugators in a group can be considered as an effective version of the conjugacy problem. The conjugacy length function provides a measure for these lengths. We study the behaviour of conjugacy length…
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 $BSA(G)$…
In a recent paper, Henry Bradford showed that all sufficiently fast growing functions appear as the residual finiteness growth function of some group. In this paper we show that the groups there constructed are conjugacy separable and that…
We prove that for any infinite right-angled Coxeter or Artin group, its spherical and geodesic growth rates (with respect to the standard generating set) either take values in the set of Perron numbers, or equal $1$. Also, we compute the…