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14.34 (1999)

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By definition, a locally compact group has the Kazhdan T-property if the trivial representation is an isolated point in the natural topological space of unitary representations of the group. Does there exist a profinite group with two dense discrete subgroups one of which is amenable, and the other has the Kazhdan T-property?

See (A. Lubotzky, Discrete groups, expanding graphs and invariant measures (Progress in Mathematics, Boston, Mass., 125), Birkhäuser, Basel, 1994) for further motivation.

Contributor: R. I. Grigorchuk, A. Lubotzky

Progress

Yes, there does (M. Kassabov, Invent. Math., 170 (2007), 297–326; M. Ershov, A. Jaikin-Zapirain, Invent. Math., 179, 303–347).

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