Quasi-automorphisms of the infinite rooted 2-edge-coloured binary tree

TL;DR

This paper investigates a class of groups called quasi-automorphisms acting on an infinite 2-edge coloured binary tree, establishing their algebraic properties, presentations, and homological invariants, and relating them to Thompson's groups.

Contribution

It introduces new groups related to Thompson's groups, proves they are of type F_infinity, and computes their presentations, subgroup structures, and homological invariants.

Findings

QF, ~QT, ~QV are of type F_infinity

Finite presentations for these groups are provided

The normal subgroup structure and rational homology are calculated

Abstract

We study the group $QV$, the self-maps of the infinite $2$-edge coloured binary tree which preserve the edge and colour relations at cofinitely many locations. We introduce related groups $QF$, $QT$, $\tilde{Q}T$, and $\tilde{Q}V$, prove that $QF$, $\tilde{Q}T$, and $\tilde{Q}V$ are of type $\mathrm{F}_\infty$, and calculate finite presentations for them. We calculate the normal subgroup structure and rational homology of all $5$ groups, the Bieri--Neumann--Strebel--Renz invariants of $QF$, and discuss the relationship of all $5$ groups with other generalisations of Thompson's groups.