Amenability versus non-exactness of dense subgroups of a compact group

TL;DR

This paper constructs dense subgroups of a compact group that exhibit both amenability and non-exactness, revealing complex interactions between these properties in group theory.

Contribution

It introduces a novel construction of dense subgroups with contrasting properties within a compact group, combining amenability and non-exactness.

Findings

Existence of dense amenable subgroups in a compact group.

Construction of non-exact dense subgroups using LEF approximations.

Demonstration of the coexistence of amenability and non-exactness in dense subgroups.

Abstract

Given a countable residually finite group, we construct a compact group K and two elements w and u of K with the following properties: The group generated by w and the cube of u is amenable, the group generated by w and u contains a copy of the given group, and these two groups are dense in K. By combining it with a construction of non-exact groups that are LEF by Osajda and Arzhantseva--Osajda and formation of diagonal products, we construct an example for which the latter dense group is non-exact. Our proof employs approximations in the space of marked groups of LEF ("Locally Embeddable into Finite groups") groups.