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Maarten Lathouwers

Publications and source records attributed to Maarten Lathouwers.

5 recordsLinked to original sources
arXiv2025

Twisted conjugacy growth series of virtually abelian groups

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 series is always an explicitly computable $\mathbb{N}$-rational function. As a corollary, we obtain a similar result for the relative growth series of a twisted conjugacy class in a virtually abelian group.

math.GR↗
arXiv2024

$R_{\infty}$-property for groups commensurable to nilpotent quotients of RAAGs

Let $G$ be a group and $φ$ an automorphism of $G$. Two elements $x,y \in G$ are said to be $φ$-conjugate if there exists a third element $z \in G$ such that $z x φ(z)^{-1} = y$. Being $φ$-conjugate defines an equivalence relation on $G$. The group $G$ is said to have the $R_{\infty}$-property if all its automorphisms $φ$ have infinitely many $φ$-conjugacy classes. For finitely generated torsion-free nilpotent groups, the so-called Mal'cev completion of the group is a useful tool in studying this property. Two groups have isomorphic Mal'cev completions if and only if they are abstractly commensurable. This raises the question whether the $R_{\infty}$-property is invariant under abstract commensurability within the class of finitely generated torsion-free nilpotent groups. We show that the answer to this question is negative and provide counterexamples within a class of 2-step nilpotent groups associated to edge-weighted graphs. These groups are commensurable to 2-step nilpotent quotients of right-angled Artin groups.

math.GR↗
arXiv2024

The twisted conjugacy growth of virtually abelian groups

In this paper, we study the asymptotics of several growth functions related to twisted conjugacy on virtually abelian groups. First, we study the twisted conjugacy growth function, which counts the number of twisted conjugacy classes intersecting the ball of radius r around the identity element. Thereafter we study the function that measures the size of the intersection of a given twisted conjugacy class with the balls around the identity element. Finally, we study the number of induced twisted conjugacy classes in certain finite quotients of the given virtually abelian group. In each of these cases we obtain a polynomial asymptotic behaviour of these growth functions.

math.GR↗
arXiv2022

The Reidemeister spectrum of 2-step nilpotent groups determined by graphs

In this paper we study the Reidemeister spectrum of 2-step nilpotent groups associated to graphs. We develop three methods, based on the structure of the graph, that can be used to determine the Reidemeister spectrum of the associated group in terms of the Reidemeister spectra of groups associated to smaller graphs. We illustrate our methods for several families of graphs, including all the groups associated to a graph with at most four vertices. We also apply our results in the context of topological fixed point theory for nilmanifolds.

math.GR↗