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Mathematics > Group Theory

arXiv:1805.01398 (math)
[Submitted on 3 May 2018 (v1), last revised 13 Mar 2019 (this version, v4)]

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

Authors:Masato Mimura
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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.
Comments: 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
Subjects: Group Theory (math.GR); Operator Algebras (math.OA)
Cite as: arXiv:1805.01398 [math.GR]
  (or arXiv:1805.01398v4 [math.GR] for this version)
  https://doi.org/10.48550/arXiv.1805.01398
Related DOI: https://doi.org/10.1112/jlms.12229

Submission history

From: Masato Mimura [view email]
[v1] Thu, 3 May 2018 16:10:29 UTC (22 KB)
[v2] Fri, 27 Jul 2018 17:31:49 UTC (33 KB)
[v3] Mon, 10 Sep 2018 06:41:37 UTC (34 KB)
[v4] Wed, 13 Mar 2019 08:44:59 UTC (37 KB)
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