A class of rearrangement groups that are not invariably generated. (English) Zbl 07922707
A group \(G\) is invariably generated if there exists a subset \(S \subseteq G\) such that, for every choice \(g_{s}\in G\) for \(s \in S\), the group \(G\) is generated by \(\{s^{g_{s}} \mid s \in S \}\). T. Gelander, G. Golan and K. Juschenko, in [J. Algebra 478, 261–270 (2017; Zbl 1390.20035)], showed that Thompson groups \(T\) and \(V\) are not invariably generated.
A generalizations of Thompson groups is the class of rearrangement groups, introduced by J. Belk and B. Forrest in [Trans. Am. Math. Soc. 372, No. 7, 4509–4552 (2019; Zbl 1480.20095)], which are defined as certain groups of homeomorphisms of limit spaces of sequences of graphs. The paper under review is devoted to the study of such groups, in particular the authors prove the following result.
Theorem 1.1. A \(CO\)-transitive subgroup \(G\) of a rearrangement group \(G_{\mathcal{X}}\) is not invariably generated.
A group \(G\) is called \(CO\)-transitive (compact-open transitive, see [S. Kim, T. Koberda, and Y. Lodha, Ann. Sci. Éc. Norm. Supér. (4) 52, No. 4, 797-820 (2019; Zbl 1516.57053)]) if \(G\) acts on a space \(X\) in such a way that, for each proper compact \(K\) and each nonempty open \(U\) of \(X\), there is an element of \(G\) that maps \(K\) inside \(U\).
As a consequence of Theorem 1.1, the authors prove that the following groups are not invariably generated: the Higman-Thompson groups \(T_{n,r}\) and \(V_{n,r}\); the Basilica Thompson group \(T_{B}\) and its generalizations; the Airplane rearrangement group \(T_{A}\); the Vicsek rearrangement group and its generalizations; topological full groups of one-sided irreducible branching edge-shifts.
A generalizations of Thompson groups is the class of rearrangement groups, introduced by J. Belk and B. Forrest in [Trans. Am. Math. Soc. 372, No. 7, 4509–4552 (2019; Zbl 1480.20095)], which are defined as certain groups of homeomorphisms of limit spaces of sequences of graphs. The paper under review is devoted to the study of such groups, in particular the authors prove the following result.
Theorem 1.1. A \(CO\)-transitive subgroup \(G\) of a rearrangement group \(G_{\mathcal{X}}\) is not invariably generated.
A group \(G\) is called \(CO\)-transitive (compact-open transitive, see [S. Kim, T. Koberda, and Y. Lodha, Ann. Sci. Éc. Norm. Supér. (4) 52, No. 4, 797-820 (2019; Zbl 1516.57053)]) if \(G\) acts on a space \(X\) in such a way that, for each proper compact \(K\) and each nonempty open \(U\) of \(X\), there is an element of \(G\) that maps \(K\) inside \(U\).
As a consequence of Theorem 1.1, the authors prove that the following groups are not invariably generated: the Higman-Thompson groups \(T_{n,r}\) and \(V_{n,r}\); the Basilica Thompson group \(T_{B}\) and its generalizations; the Airplane rearrangement group \(T_{A}\); the Vicsek rearrangement group and its generalizations; topological full groups of one-sided irreducible branching edge-shifts.
Reviewer: Enrico Jabara (Venezia)
MSC:
| 20F65 | Geometric group theory |
| 20F05 | Generators, relations, and presentations of groups |
| 20E45 | Conjugacy classes for groups |
| 57S05 | Topological properties of groups of homeomorphisms or diffeomorphisms |
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