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I know that the Higman group is the most widely studied candidate right now, but what are the others? For example, is (are) Thompson's group(s) sofic? And what about the Burger-Mozes groups? I haven't found anything about their soficity, are they believed (or proved) to be sofic?

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    $\begingroup$ All non-amenable (or, unknown to be amenable) isolated groups (see the various examples in arxiv.org/abs/math/0511714) are natural candidates. This includes non-amenable finitely presented simple groups such as Burger-Mozes groups or Thompson's groups of the circle or Cantor set, the Thompson group of the interval. Also this includes some non-residually finite lattices, see 5.8 in the same reference. These ones are really intriguing for soficity because the failure of residual finiteness lies on only one element (say, when the finite residual has order 2). $\endgroup$
    – YCor
    Sep 3, 2016 at 19:11
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    $\begingroup$ @YCor, your comment seems like a great answer to the question! I suggest you post it as an answer. $\endgroup$
    – HJRW
    Sep 5, 2016 at 15:16
  • $\begingroup$ @YCor could you post an answer with what you know about soficity of non-amenable isolated groups? I think it deserves to be pointed out that you showed in arxiv.org/abs/0906.3374 that one of these natural candidates is sofic. $\endgroup$
    – Giles Gardam
    Dec 21, 2021 at 13:47

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