# Coloured Neretin Groups

## TL;DR

This paper explores the structure of certain tree automorphism groups, establishing conditions for their subgroups to resemble topological full groups, and demonstrating properties like compact generation, virtual simplicity, and the absence of lattices.

## Contribution

It provides new criteria for subgroup isomorphism to topological full groups and applies these to universal groups, revealing their compact generation and simplicity properties.

## Key findings

- Almost automorphism groups are compactly generated.
- Some groups are virtually simple.
- Certain groups have no lattices.

## Abstract

We give sufficient conditions for a subgroup of a tree almost automorphism group to be isomorphic to the topological full groups of a one-sided shift in the sense of Matui. As an application, we show that almost automorphism groups of trees obtained from universal groups constructed by Burger and Mozes are compactly generated and virtually simple. In addition, using the approach of Bader, Caprace, Gelander and Mozes we show that some of these almost automorphism groups do not have any lattice.