# Quasi-Automorphism Groups of Type F-infinity

## TL;DR

This paper proves that the quasi-automorphism groups of the infinite binary tree, including QF, QT, and QV, have type F-infinity, using hybrid diagrams and actions on CAT(0) cubical complexes.

## Contribution

It introduces a novel use of hybrid diagrams to establish the finiteness properties of these quasi-automorphism groups.

## Key findings

- All groups studied have type F-infinity.
- Hybrid diagrams share properties with planar and braided diagrams.
- Groups act properly on CAT(0) cubical complexes.

## Abstract

The groups QF, QT, and QV are groups of quasi-automorphisms of the infinite binary tree. Their names indicate a similarity with Thompson's well-known groups F, T, and V. We will use the theory of diagram groups over semigroup presentations to prove that all of the above groups (and several generalizations) have type F-infinity. Our proof uses certain types of hybrid diagrams, which have properties in common with both planar diagrams and braided diagrams. The diagram groups defined by hybrid diagrams also act properly and isometrically on CAT(0) cubical complexes.