$\DeclareMathOperator\FSym{FSym}\DeclareMathOperator\Sym{Sym}$*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 $G=\mathbf{Z}$)?



Permutations of a group that are eventually left translations
**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.**](http://dx.doi.org/10.1515/JGT.2006.011), J. Group Theory 9, No. 2, 161-171 (2006). [ZBL1153.20040](https://zbmath.org/?q=an:1153.20040) [arXiv:math/0305352](https://arxiv.org/abs/math/0305352).


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


