The $R_\infty$ property for crystallographic group of Sol

TL;DR

This paper investigates the Reidemeister number for automorphisms of crystallographic groups of Sol, showing that most such groups possess the $R_ty$ property, indicating infinite Reidemeister numbers.

Contribution

It establishes the $R_ty$ property for most crystallographic groups of Sol using the averaging formula, expanding understanding of Reidemeister numbers in geometric group theory.

Findings

Most crystallographic groups of Sol have the $R_ty$ property.

The averaging formula is effective in analyzing Reidemeister numbers.

The study covers all nine types of crystallographic groups of Sol.

Abstract

There are 9 kinds of crystallographic groups $\Pi$ of Sol. For any automorphism $\varphi$ on $\Pi$, we study the Reidemeister number $R(\varphi)$. Using the averaging formula for the Reidemeister numbers, we prove that most of the crystallographic groups of Sol have the $R_\infty$ property.