# About von Neumann's problem for locally compact groups

## TL;DR

This paper generalizes geometric solutions to von Neumann's problem for locally compact groups, linking geometric properties with algebraic features of wobbling groups and strengthening existing results on paradoxical decompositions.

## Contribution

It introduces a broader framework for understanding paradoxical decompositions in locally compact groups using Borel and clopen translations, extending prior work.

## Key findings

- Established a connection between coarse space geometry and wobbling group algebraic properties.
- Generalized Whyte's solution to von Neumann's problem for a wider class of groups.
- Strengthened Paterson's results on Borel paradoxical decompositions.

## Abstract

We note a generalization of Whyte's geometric solution to the von Neumann problem for locally compact groups in terms of Borel and clopen piecewise translations. This strengthens a result of Paterson on the existence of Borel paradoxical decompositions for non-amenable locally compact groups. Along the way, we study the connection between some geometric properties of coarse spaces and certain algebraic characteristics of their wobbling groups.