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

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 u3 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 locally embeddable into finite (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 groups.