# Finiteness properties of totally disconnected locally compact groups

## TL;DR

This paper explores finiteness properties of totally disconnected locally compact groups over various rings, establishing analogues of classical criteria, invariance under quasi-isometry, and introducing graph-wreath products with their finiteness characteristics.

## Contribution

It extends finiteness property theory to totally disconnected locally compact groups, including new criteria, invariance results, and the concept of graph-wreath products.

## Key findings

- Finiteness properties satisfy analogues of classical results for discrete groups.
- FP_n and F_n properties are quasi-isometric invariants.
- Introduction of graph-wreath products and analysis of their finiteness properties.

## Abstract

In this paper we investigate finiteness properties of totally disconnected locally compact groups for general commutative rings $R$, in particular for $R = \mathbb{Z}$ and $R= \mathbb{Q}$. We show these properties satisfy many analogous results to the case of discrete groups, and we provide analogues of the famous Bieri's and Brown's criteria for finiteness properties and deduce that both $FP_n$-properties and $F_n$-properties are quasi-isometric invariant. Moreover, we introduce graph-wreath products in the category of totally disconnected locally compact groups and discuss their finiteness properties.