On geometric aspects of diffuse groups

TL;DR

This paper explores the geometric properties of diffuse groups, providing examples, non-examples, and addressing open questions about their orderability and relation to fundamental groups of manifolds.

Contribution

It extends Bowditch's concept of diffuse groups by analyzing geometric examples and non-examples, including fundamental groups of flat and hyperbolic manifolds, and answers an open question on orderability.

Findings

Identified groups that are diffuse but not left-orderable

Provided geometric examples and non-examples of diffuse groups

Addressed an open question in the theory of diffuse groups

Abstract

Bowditch introduced the notion of diffuse groups as a geometric variation of the unique product property. We elaborate on various examples and non-examples, keeping the geometric point of view from Bowditch's paper. In particular, we discuss fundamental groups of flat and hyperbolic manifolds. The appendix settles an open question by providing an example of a group which is diffuse but not left-orderable.