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

TL;DR

This paper proves that Reidemeister zeta functions of automorphisms of certain low-dimensional almost-crystallographic groups are rational, extending previous results to groups up to dimension 3.

Contribution

It establishes the rationality of Reidemeister zeta functions for automorphisms of low-dimensional almost-crystallographic groups, specifically those with diagonal holonomy rac{1}{2}.

Findings

Reidemeister zeta functions are rational for automorphisms of crystallographic groups with diagonal holonomy rac{1}{2}.

Reidemeister zeta functions of automorphisms of almost-crystallographic groups up to dimension 3 are rational.

The result extends the class of groups known to have rational Reidemeister zeta functions.

Abstract

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.