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In 1993, Lubotzky and Weiss conjectured that if a compact group admits twofinitely generated dense subgroups, one of which is amenable and the other hasKazhdan's property (T), then it would be finite. This conjecture was resolvedin the negative by Ershov and Jaikin-Zapirain, and by Kassabov around 2010. Inthe present paper, we provide an extreme counterexample to this conjecture.More precisely, the latter dense group with property (T) may contain a givencountable residually finite group; in particular, it can be non-exact by aresult of Osajda. We may construct these counterexamples with a compact groupcommon for all countable residually finite groups.

An extreme counterexample to the Lubotzky--Weiss conjecture