Centralizers in the R. Thompson group $V_n$

Abstract

Let n 2 and let˛2 V n be an element in the Higman-Thompson group V n. We study the structure of the centralizer of˛2 V n through a careful analysis of the action of h˛i on the Cantor set C. We make use of revealing tree pairs as developed by Brin and Salazar from which we derive discrete train tracks and flow graphs to assist us in our analysis. A consequence of our structure theorem is that element centralizers are finitely generated. Along the way we give a short argument using revealing tree pairs which shows that cyclic groups are undistorted in V n .

Embeddings into Thompson's group V and coCF groups

“…The essence of the argument below will be clear to any reader who has digested the material on revealing pairs for elements of V (for instance, as presented in [5], which has an expository section written to explain these objects). Note that revealing pairs are introduced by Brin in [7], and that Higman in [14] had already developed an analogous technology.…”

“…For example, a possibility for m is the least common multiple of the set of lengths of all finite periodic orbits, as there are only finitely many such lengths. Now, the element α admits a finite set of points I(α), which we will call the important points of α (following [5]), consisting of the repelling and attracting points in the Cantor set under the action of α . For each point in the set I(α), it is the case that α restricted to some small interval U p containing p is an affine map which fixes exactly the point p, where the slope of this map is 2 sp , for some s p a fixed non-zero integer.…”

“…One can think of the result as a total "speed" along an orbit, and it is useful in many ways. For instance, this product-of-slopes calculation for an orbit is used in section 7 of [5] to analyse element centralisers. As described later in this section, it is also used in [5] to show that all cyclic subgroups are undistorted in V .…”

“…First, recall Theorem 1.3 in [5] which says that all cyclic groups are undistorted in V . Now recall that when |m| = |n|, the group BS(m, n) has distorted cyclic subgroups.…”

Dynamique des échanges d’intervalles des groupes de Higman-Thompson V_{r,m}

“…Plus précisément, nous montrons que tout élément de V r,m a ses orbites propres et possède au moins une orbite périodique ou un cycle périodique. Nous avons récemment appris qu'un résultat analogue se trouve dans le travail de [3]. La preuve qui y est donnée est "combinatoire" (basée sur la représentation d'un élément de V r,m comme classe d'équivalence de paire d'arbres).…”