arXiv subjects

Sam Tertooy

Publications and source records attributed to Sam Tertooy.

12 recordsLinked to original sources

arXiv2026

A Hirsch length inequality

Let $H$ and $K$ be subgroups of a virtually polycyclic group $G$. We prove the Hirsch length inequality $$h(H)+h(K) \leq h(H\cap K)+h(G).$$ We show that equality holds when the number of $(H,K)$-double cosets is finite, and that the converse holds when $G$ is nilpotent. We also apply this to twisted conjugacy, showing that for homomorphisms $\varphi,\psi \colon G \to H$ with $G$ and $H$ virtually polycyclic, there is a connection between the Hirsch lengths of $G$, $H$, and the coincidence subgroup $\mathrm{Coin}(\varphi,\psi)$, and the finiteness of the Reidemeister number $R(\varphi,\psi)$.

math.GR↗

arXiv2026

Class numbers and nilpotent subgroups of $\mathrm{PGL}(2,q)$

We show that for certain odd prime powers $q$, the number of conjugacy classes of $\mathrm{PGL}(2,q)$ is greater than the order of its largest nilpotent subgroup. This answers negatively a question of Liebeck and Pyber.

math.GR↗

arXiv2026

Bi-twisted conjugacy in finite groups

We provide two alternative ways to determine the number of bi-twisted conjugacy classes in a finite group: one using irreducible characters and one using ordinary conjugacy classes. In addition, we show various equalities and (sharp) inequalities for Reidemeister numbers, as well as relations between bi-twisted conjugacy, representation theory, and fixed-point free automorphisms.

math.GR↗

arXiv2025

Algorithms for twisted conjugacy classes of polycyclic-by-finite groups II

We construct an algorithm that, given a pair of homomorphisms between polycyclic-by-finite groups, determines whether their Reidemeister number is finite, and if so returns a set of representatives of the twisted conjugacy classes. Moreover, we show how this algorithm can be applied to compute double cosets and orbits of affine actions.

math.GR↗

arXiv2023

Extreme Reidemeister spectra of finite groups

We extend the notions of "$R_\infty$-property" and "full (extended) Reidemeister spectrum" to finite groups in a meaningful way. We provide examples of finite groups admitting these properties, if they exist, by looking at groups of small order as well as (quasi)simple groups.

math.GR↗

arXiv2023

Twisted conjugacy and separability

A group $G$ is twisted conjugacy separable if for every automorphism $\varphi$, distinct $\varphi$-twisted conjugacy classes can be separated in a finite quotient. Likewise, $G$ is completely twisted conjugacy separable if for any group $H$ and any two homomorphisms $\varphi,\psi$ from $H$ to $G$, distinct $(\varphi,\psi)$-twisted conjugacy classes can be separated in a finite quotient. We study how these properties behave with respect to taking subgroups, quotients and finite extensions, and compare them to other notions of separability in groups. Finally, we show that for polycyclic-by-nilpotent-by-finite groups, being completely twisted conjugacy separable is equivalent to all quotients being residually finite.

math.GR↗

arXiv2021

The Reidemeister spectra of low dimensional crystallographic groups

In this paper we study the number of twisted conjugacy classes (the Reidemeister number) for automorphisms of crystallographic groups. We present two main algorithms for crystallographic groups whose holonomy group has finite normaliser in $\operatorname{GL}_n(\mathbb{Z})$. The first algorithm calculates whether a group has the $R_\infty$-property; the second calculates the Reidemeister spectrum. We apply these algorithms to crystallographic groups up to dimension $6$.

math.GR↗

arXiv2021

Algorithms for twisted conjugacy classes of polycyclic-by-finite groups

We construct two practical algorithms for twisted conjugacy classes of polycyclic-by-finite groups. The first algorithm determines whether two elements of a group are twisted conjugate for two given endomorphisms, under the condition that the Reidemeister coincidence number of these endomorphisms is finite. The second algorithm determines representatives of the Reidemeister coincidence classes of two endomorphisms if their Reidemeister coincidence number is finite, or returns "fail" if the Reidemeister coincidence number is infinite.

math.GR↗

arXiv2021

Reidemeister zeta functions of low-dimensional almost-crystallographic groups are rational

We prove that the Reidemeister zeta functions of automorphisms of crystallographic groups with diagonal holonomy $\mathbb{Z}_2$ are rational. As a result, we obtain that Reidemeister zeta functions of automorphisms of almost-crystallographic groups up to dimension 3 are rational.

math.GR↗

arXiv2021

Reidemeister spectra for solvmanifolds in low dimensions

The Reidemeister number of an endomorphism of a group is the number of twisted conjugacy classes determined by that endomorphism. The collection of all Reidemeister numbers of all automorphisms of a group $G$ is called the Reidemeister spectrum of $G$. In this paper, we determine the Reidemeister spectra of all fundamental groups of solvmanifolds up to Hirsch length 4.

math.GR↗

arXiv2021

Fixed points of diffeomorphisms on nilmanifolds with a free nilpotent fundamental group

Let $M$ be a nilmanifold with a fundamental group which is free $2$-step nilpotent on at least 4 generators. We will show that for any nonnegative integer $n$ there exists a self-diffeomorphism $h_n$ of $M$ such that $h_n$ has exactly $n$ fixed points and any self-map $f$ of $M$ which is homotopic to $h_n$ has at least $n$ fixed points. We will also shed some light on the situation for less generators and also for higher nilpotency classes.

math.AT↗

arXiv2019

The Reidemeister spectrum of low dimensional almost-crystallographic groups

We determine which non-crystallographic, almost-crystallographic groups of dimension 4 have the $R_\infty$-property. We then calculate the Reidemeister spectra of the 3-dimensional almost-crystallographic groups and the 4-dimensional almost-Bieberbach groups.

math.GR↗