Commutativity and free products in Thompson's group V

Abstract

We broaden the theory of dynamical interpretation, investigate the property of commutativity and explore the subject of subgroups forming free products in Thompson's group V.

We expand Brin's terminology for a revealing pair to an any tree pair. We use it to analyse the dynamical behaviour of an arbitrary tree pair which cannot occur in a revealing pair. Hence, we design a series of algorithms generating Brin's revealing pair from any tree pair, by successively eliminating the undesirable structures. To detect patterns and transitioning between tree pairs, we introduce a new combinatorial object called the chains graph. A newly defined, unique and symmetrical type of a tree pair, called a balanced tree pair, stems from the use of the chains graphs.

The main theorem of Bleak et al. in "Centralizers in the R. Thompson's Group V_n" states the necessary structure of the centraliser of an element of V. We provide a converse to this theorem, by proving that each of the predicted structures is realisable. Hence we obtain a complete classification of centralisers in V. We give an explicit construction of an element of V with prescribed centraliser. The underlying concept is to embed a Cayley graph of a finite group into the flow graph (introduced in Bleak et al.) of the desired element. To reflect the symmetry, we present the resulting element in terms of a balanced tree pair.

The group V is conjectured to be a universal coCF group, which generates interest in studying its subgroups. We develop a better understanding of embeddings into V by providing a necessary and sufficient dynamical condition for two subgroups (not both torsion) to form a free product in V. For this, we use the properties, explored in Bleak and Salazar-Díaz "Free Products in Thompson's Group V", of sets of so--called important points, and the Ping-Pong action induced on them.