Related papers: Subgroup growth in free class-$2$-nilpotent groups
We compute in a direct (not algorithmic) way the zeta function of all supersingular curves of genus 2 over a finite field k, with many geometric automorphisms. We display these computations in an appendix where we select a family of…
We study the product formula for Reidemeister numbers on finitely generated torsion-free nilpotent groups in two ways. On the one hand, we generalise the product formula to central extensions. On the other hand, we derive general results…
The zeta function of an integral lattice $\Lambda$ is the generating function $\zeta_{\Lambda}(s) = \sum\limits_{n=0}^{\infty} a_n n^{-s}$, whose coefficients count the number of left ideals of $\Lambda$ of index $n$. We derive a formula…
We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be…
We consider coincidence Reidemeister zeta functions for tame endomorphism pairs of nilpotent groups of finite rank, shedding new light on the subject by means of profinite completion techniques. In particular, we provide a closed formula…
We introduce and study the Hilbert space of $(L^2,\Gamma,\chi)$-likewise theta functions on $\mathbb{R}^d$ with respect to a given discrete subgroup $\Gamma$ of arbitrary rank and a character $\chi$ of $\Gamma$. A concrete description is…
We consider expansive group actions on a compact metric space containing a special fixed point denoted by $0$, and endomorphisms of such systems whose forward trajectories are attracted toward $0$. Such endomorphisms are called…
This is the first paper in a series of three where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Our main goal is to show that group actions on Z^n-trees give one a powerful tool to…
We give sufficient conditions for when groups generated by automata in a class $\mathcal{C}$ of transducers, which contains the class of reset automata transducers, have infinite order. As a consequence we also demonstrate that if a group…
We give a "limit-free formula" simplifying the computation of the topological entropy for topological automorphisms of totally disconnected locally compact groups. This result allows us to extend several basic properties of the topological…
The study of subshifts on groups different from $\mathbb{Z}$, such as $\mathbb{Z}^d$, $d\geq 2$, has been a subject of intense research in recent years. These investigations have unveiled aremarkable connection between dynamics and…
Suppose $Y$ is a regular covering of a graph $X$ with covering transformation group $\pi = \mathbb{Z}$. This paper gives an explicit formula for the $L^2$ zeta function of $Y$ and computes examples. When $\pi = \mathbb{Z}$, the $L^2$ zeta…
We present a structural description of finite nilpotent groups of class at most $2$ using a specified number of subdirect and central products of $2$-generated such groups. As a corollary, we show that all of these groups are isomorphic to…
We consider semigroups of Ruelle-expanding maps, parameterized by random walks on the free semigroup, with the aim of examining their complexity and exploring the relation between intrinsic properties of the semigroup action and the…
We compute the representation zeta functions of some finitely generated nilpotent groups associated to unipotent group schemes over rings of integers in number fields. These group schemes are defined by Lie lattices whose presentations are…
Motivated by the connection to the pair correlation of the Riemann zeros, we investigate the second derivative of the logarithm of the Riemann zeta function, in particular the zeros of this function. Theorem 1 gives a zero-free region.…
We define zeta functions for the adjoint action of GL(n) on its Lie algebra and study their analytic properties. For n<4 we are able to fully analyse these functions, and recover the Shintani zeta function for the prehomogeneous vector…
In this paper we continue to study the Reidemeister zeta function. We prove P\'olya -- Carlson dichotomy between rationality and a natural boundary for analytic behavior of the Reidemeister zeta function for a large class of automorphisms…
A beautifully simple free generating set for the commutator subgroup of a free group was constructed by Tomaszewski. We give a new geometric proof of his theorem, and show how to give a similar free generating set for the commutator…
We give a subexponential upper bound and a superpolynomial lower bound on the growth function of the Fabrykowski-Gupta group. As a consequence, we answer negatively a question by Longobardi, Maj and Rhemtulla about characterizing groups…