Vol. 287, No. 2, 2017

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Some closure results for $\mathcal{C}$-approximable groups

Derek F. Holt and Sarah Rees

Vol. 287 (2017), No. 2, 393–409

We investigate closure results for C-approximable groups, for certain classes C, of groups with invariant length functions. In particular we prove, each time for certain (but not necessarily the same) classes C that: (i) the direct product of two C-approximable groups is C-approximable; (ii) the restricted standard wreath product G H is C-approximable when G is C-approximable and H is residually finite; and (iii) a group G with normal subgroup N is C-approximable when N is C-approximable and GN is amenable. Our direct product result is valid for LEF, weakly sofic and hyperlinear groups, as well as for all groups that are approximable by finite groups equipped with commutator-contractive invariant length functions (considered by A.Thom). Our wreath product result is valid for weakly sofic groups, and we prove it separately for sofic groups. This last result has recently been generalised by Hayes and Sale, who proved that the restricted standard wreath product of any two sofic groups is sofic. Our result on extensions by amenable groups is valid for weakly sofic groups, and was proved by Elek and Szabó (2006) for sofic groups N.

C-approximable group, sofic, hyperlinear, weakly sofic, linear sofic
Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20E22
Received: 8 January 2016
Revised: 25 September 2016
Accepted: 26 September 2016
Published: 9 March 2017
Derek F. Holt
Mathematics Institute
University of Warwick
United Kingdom
Sarah Rees
School of Mathematics and Statistics
University of Newcastle
United Kingdom