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

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 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.