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 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.

32 pages, explanation brushed up; 28 pages, can take $t=u^3$ in the main theorem (Theorem A); 27 pages, results are optimized; 19 pages, no figure