@article{9db7fc27b2b343a093c884819c25f16b,
title = "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.",
keywords = "20D06 (secondary), 20F69 (primary)",
author = "Masato Mimura",
note = "Funding Information: This work has been done during the two-year stay of the author in the {\'E}cole Polytechnique F{\'e}d{\'e}rale de Lausanne supported by Grant-in-Aid for JSPS Oversea Research Fellowships, as well as his visit to University of Bristol during that stay. The author wishes to express his gratitude to Professor Nicolas Monod and Mrs. Marcia Gouffon at the EPFL and Professor John Mackay at the University of Bristol for their hospitality and help, respectively, on his stay and visit. The author thanks Yash Lodha and Michael Magee for discussion, respectively, on Hall's embedding theorem and on Goursat's lemma. He is grateful to Goulnara Arzhantseva, Yoshikata Kida, and Sven Raum for several comments and suggestions, Piotr W. Nowak for the reference, Damian Osajda for discussion on his construction in, Damian Sawicki for comments and the reference, and Ryokichi Tanaka for discussions on LFNF-lifts. He is grateful to the anonymous referee for the reference. His/her comments have considerably enhanced the readability of the present paper. Publisher Copyright: {\textcopyright} 2019 The Author(s). The Journal of the London Mathematical Society is copyright {\textcopyright} London Mathematical Society",
year = "2019",
month = oct,
day = "1",
doi = "10.1112/jlms.12229",
language = "English",
volume = "100",
pages = "592--622",
journal = "Journal of the London Mathematical Society",
issn = "0024-6107",
publisher = "Oxford University Press",
number = "2",
}