Conjugacy and centralisers in Thompson's group T

Abstract

The Thompson family of groups F < T < V are well-known as interesting (counter-) examples in group theory. Working algebraically with these objects is difficult, and yet the groups are computationally tractable. For instance, T and V are infinite simple groups; despite this, both are finitely presented. This thesis studies the middle group T, using the piecewise-linear function point of view. We present a solution to the conjugacy problem in this group, adapting the approach of Kassabov and Matucci1 to 1 Kassabov and Matucci 2012. the same problem in F. Conjugacy of elements in T was shown2 to 2 Belk and Matucci 2014. be decidable by Belk and Matucci; however our approach constructs explicit conjugators (when they exist). Later, we refine the description given by Matucci3 of nontorsion elements’ centralisers in T. 3 Matucci 2008, Chapter 7. * * * The first chapter introduces the world of Thompson’s groups. The sections on cyclic order, the generalised groups PLS,G and groupoid PL2, and on the Cantor space are particularly important for readers interested in the rest of the thesis. The second chapter discusses Thompson’s groups from a dynamical point of view. We summarise how F, T and V rerrange the interval, noting the distinction between dyadic and nondyadic points. Focussing on T, we introduce the rotation number and explain what we can learn from it. Amidst all this we present a number of intermediate results, forming a toolkit for use in later chapters. The third chapter studies conjugacy in T. We narrow the search space by finding constraints that a conjugator must satisfy. Next, we break the conjugacy problem into a search for a coarse and fine conjugator, the product of which—if they exist—is a bona fide conjugator. We solve these search problems,4 and thus solve the conjugacy problem in T. 4 Lemma 3.3.2 and Algorithms 3.3.5 and 3.4.8. In the fourth chapter, we study element centralisers in T via a particular group extension. We focus on nontorsion elements, providing small details missing from Matucci’s proof which identifies the extension’s kernel.5 We explain how to find the size of the extension’s quotient, by 5 Theorem 4.2.1 and Remark 4.2.2. reducing the problem to a search for coarse conjugators.6 6 Algorithm 4.2.3. The final chapter describes the extension structure of CT(α) in more detail.7 We do so by classifying α into one of four cases. In all but 7 Section 5.2. one case, this extension splits (as a wreath or direct product); in the remaining case, we identify8 exactly when the extension splits (again as 8 See Proposition 5.2.15 and the summary in in Theorem 5.3.2. a wreath product). In each case, we describe the centraliser’s structure9 9 Proposition 5.2.17. in terms of integer parameters. We then show how to construct10 an 10 Corollaries 5.2.6, 5.2.16, 5.2.18, 5.2.23 element of T whose centraliser has a given list of parameters.