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$\DeclareMathOperator\FSym{FSym}\DeclareMathOperator\Sym{Sym}$Notation: for $X$ a set, $\Sym(X)$ the group of permutations of $X$, and let $\FSym(X)$ be the subgroup of finitely supported permutations of $X$ (it is generated by transpositions).

Let $G$ be an infinite group. Let $P_G$ be the subgroup of $\Sym(G)$ generated by left translations and $\FSym(G)$. Equivalently, these are permutations of $G$ that coincide with a left translation outside a finite subset.

Thus $P_G$ is naturally a semidirect product $\FSym(G)\rtimes G$. One can prove various things about this group (e.g., if $G$ is finitely generated so is $P_G$, the group $P_G$ is never finitely presented, never Kazhdan, etc.

The case $G=\mathbf{Z}$ of this construction is particularly well-known (as far as I know it essentially appears in a 1937 paper of B.H. Neumann) and well-documented.

I remember reading a paper about groups $P_G$ (at least 10 years ago), including these results (or so of them). Despite significant efforts to find the right keywords, I wasn't able to locate it. Does anybody identify this (or any paper referring to this construction in general, not just $G=\mathbf{Z}$)?

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  • $\begingroup$ The only article I know is arxiv:1503.04977, but it's not what you are looking for... $\endgroup$
    – AGenevois
    Commented Nov 18, 2021 at 9:37
  • $\begingroup$ This construction is used by Elek and Szabo to give a sofic group that is not residually amenable, but that's not news to you arxiv.org/abs/math/0305352 $\endgroup$
    – Giles Gardam
    Commented Dec 1, 2021 at 6:49
  • $\begingroup$ @GilesGardam thanks! this is very likely to be the paper I was looking for (since indeed I knew this paper, and I had forgotten that it includes this very construction). $\endgroup$
    – YCor
    Commented Dec 1, 2021 at 9:13

1 Answer 1

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Theorem (Elek–Szabo). Let $G$ be an infinite residually finite hyperbolic group with Property (T). Then $P_G$ is a finitely generated sofic group that is not residually amenable.

That's Theorem 3 of Elek, Gábor; Szabó, Endre, On sofic groups., J. Group Theory 9, No. 2, 161-171 (2006). ZBL1153.20040 arXiv:math/0305352.

The construction is also somewhat reminiscent of Houghton's groups.

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