Commutativity and free products in Thompson's group V
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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.
Thesis, PhD Doctor of Philosophy
Attribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/
Description of related resources'Centralizers in the R. Thompson's group V_n' (2013) / Collin Bleak, Hannah Bowman, Alison Gordon Lynch, Garrett Graham, Jacob Hughes, Francesco Matucci, Eugenia Sapir, Groups Geom. Dyn. 7, no. 4, 821-865
'Free products in R. Thompson's group V' (2013) / Collin Bleak and Olga Salazar-Díaz, trans. Amer. Math. Soc. 365, no. 11, 5967-5997
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