Thompson-like groups, Reidemeister numbers, and fixed points

TL;DR

This paper studies fixed-point properties of automorphisms in Thompson-like groups, providing a cohomological criterion for property R_infinity and applying it to various groups related to Thompson's F.

Contribution

It introduces a new cohomological criterion for detecting infinite fixed points in abelianizations of Thompson-like groups, expanding understanding of their automorphism fixed-point properties.

Findings

The criterion detects property R_infinity in several Thompson-like groups.

Many groups including Stein's F_{2,3} and Lodha-Moore groups satisfy R_infinity.

The method unifies fixed-point analysis across different Thompson-like groups.

Abstract

We investigate fixed-point properties of automorphisms of groups similar to R. Thompson's group $F$. Revisiting work of Gon\c{c}alves-Kochloukova, we deduce a cohomological criterion to detect infinite fixed-point sets in the abelianization, implying the so-called property $R_\infty$. Using the BNS $\Sigma$-invariant and drawing from works of Gon\c{c}alves-Sankaran-Strebel and Zaremsky, we show that our tool applies to many $F$-like groups, including Stein's $F_{2,3}$, Cleary's $F_\tau$, the Lodha-Moore groups, and the braided version of $F$.