7
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Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the permutation of $\mathbb{Z}$ which interchanges $r_1+km_1$ and $r_2+km_2$ for every $k \in \mathbb{Z}$ and which fixes everything else.

Question: Does the group ${\rm CT}(\mathbb{Z})$ generated by the set of all class transpositions have finitely generated infinite periodic subgroups?

Remark: By means of computation one can find "almost-examples" of such groups. For example let $G := \langle a, b, c, d, e \rangle$, where $a = \tau_{0(4),2(4)}$, $b = \tau_{0(5),3(5)}$, $c = \tau_{1(3),5(6)}$, $d = \tau_{1(6),3(6)}$ and $e = \tau_{2(6),4(6)}$. Then $G$ is infinite, but the subgroups generated by any $4$ of the generators are finite, and also all products of at most $6$ generators have finite order. Though unfortunately there are products of $7$ generators which happen to have infinite order (example: $(ab)^2ced$), such that $G$ is not periodic.

Added on May 15, 2018: This question has appeared as Problem 19.46 in:

Kourovka Notebook: Unsolved Problems in Group Theory. Editors V. D. Mazurov, E. I. Khukhro. 19th Edition, Novosibirsk 2018.

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  • $\begingroup$ Remark: this group a topological-full group, namely on the space $\widehat{\mathbf{Z}}$ (profinite completion of the group of integers), of the groupoid induced by the partial multiplicative action of positive rationals. $\endgroup$ – YCor May 15 '18 at 20:54
  • $\begingroup$ @YCor: As I don't know anything about topological full groups -- do you think that this is a 'substantial' characterization in the sense that one can plausibly exploit it to investigate CT($\mathbb{Z}$) and to obtain results which wouldn't be easy to obtain otherwise? $\endgroup$ – Stefan Kohl May 16 '18 at 14:51
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    $\begingroup$ Maybe more a way to anticipate on results, which would then be easy to obtain otherwise? indeed it yields an intuition. Simplicity of the derived subgroup holds in a very general context (see Nekrashevych, Groups of dynamical origin), but you know it and actually know bare simplicity, if I remember correctly. Also Matte Bon constructed periodic (infinite f.g.) subgroups in suitable full topological groups, and this was generalized by Nekrashevych (Palindromic subshifts...), maybe this can help, or not, I don't know. $\endgroup$ – YCor May 16 '18 at 21:44
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    $\begingroup$ There are topological-full groups, naturally acting on $\mathbf{Z}$, in which the set-wise stabilizer of $\mathbf{N}$ have indeed been studied and are interesting. These are indeed topological full-groups, namely restricting to partial affine maps that preserve $\mathbf{N}$. $\endgroup$ – YCor Oct 9 '18 at 23:29
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    $\begingroup$ The most studied have been topological-full groups of a minimal self-homeomorphism of the Cantor set on the one hand, and, in the groupoid setting, the Thompson-Higman groups ($V$ and generalizations). $\endgroup$ – YCor Oct 9 '18 at 23:34

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