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Centralizers in the R. Thompson group \(V_n\). (English) Zbl 1298.20052

The Higman-Thompson groups \(V_n\), with \(n\geq 2\), are generalizations of Thompson’s group \(V\). The group \(V_n\) can be defined as a subgroup of the group of homeomorphisms of the boundary of a rooted tree \(T_n\) of degree \(n\). If we fix an ordering on the children of every node in \(T_n\), an element of \(V_n\) can be described by a pair of rooted finite \(n\)-ary subtrees \(A,B\subset T_n\) together with a bijection \(\varphi\) of their sets of leaves. Such a bijection induces a unique order-preserving isomorphism between the complements of \(A\) and \(B\), which in turn yields a homeomorphism of \(\partial T_n\).
The main result of the paper is a detailed description of the centralizer of an arbitrary element of \(V_n\) as a finite direct product of semidirect products \(K_i\rtimes G_i\) and wreath products of the form \((A_j\rtimes\mathbb Z)\wr P_j\), where \(K_i\), \(G_i\), and \(P_j\) are given explicitly, and \(A_j\) are certain finite groups. The authors leave open two questions about this decomposition, which amount to asking whether the groups \(A_j\rtimes\mathbb Z\) are Abelian.
For calculations in \(V_n\) the authors use revealing pairs of M. G. Brin [Geom. Dedicata 108, 163-192 (2004; Zbl 1136.20025)]. They are pairs of trees representing an element of \(V_n\) in a way that allows to read its dynamics. As an important tool in their analysis, they introduce the notions of a discrete train track and flow graph, which model the dynamics of a homeomorphism of a totally disconnected space, allowing a level of intuitive understanding of the action of \(V_n\) on \(\partial T_n\).
It follows from the authors’ description that centralizers in \(V_n\) are finitely generated. Moreover, using the ideas developed in the paper, they give a simple proof that elements of infinite order in \(V_n\) are undistorted.

MSC:

20F65 Geometric group theory
20E07 Subgroup theorems; subgroup growth
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37E05 Dynamical systems involving maps of the interval
20E32 Simple groups
20F05 Generators, relations, and presentations of groups
57S25 Groups acting on specific manifolds

Citations:

Zbl 1136.20025
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References:

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