We give a unified solution to the conjugacy problem for Thompson's groups F, T , and V . The solution uses "strand diagrams", which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompson's groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompson's groups, and we use this correspondence to investigate the dynamics of elements of F. Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results.
Guba and Sapir asked if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson's group F . We give a solution to the latter question using elementary techniques which rely purely on the description of F as the group of piecewise linear orientationpreserving homeomorphisms of the unit interval. The techniques we develop extend the ones used by Brin and Squier allowing us to compute roots and centralizers as well. Moreover, these techniques can be generalized to solve the same question in larger groups of piecewise-linear homeomorphisms.2010 Mathematics Subject Classification. primary 20F10; secondary 20E45, 37E05.
Abstract. Lehnert and Schweitzer show in [20] that R. Thompson's group V is a co-context-free (coCF ) group, thus implying that all of its finitely generated subgroups are also coCF groups. Also, Lehnert shows in his thesis that V embeds inside the coCF group QAut(T 2,c ), which is a group of particular bijections on the vertices of an infinite binary 2-edge-colored tree, and he conjectures that QAut(T 2,c ) is a universal coCF group. We show that QAut(T 2,c ) embeds into V , and thus obtain a new form for Lehnert's conjecture. Following up on these ideas, we begin work to build a representation theory into R. Thompson's group V . In particular we classify precisely which BaumslagSolitar groups embed into V .
Abstract. We prove that Claas Röver's Thompson-Grigorchuk simple group V G has type F∞. The proof involves constructing two complexes on which V G acts: a simplicial complex analogous to the Stein complex for V , and a polysimiplical complex analogous to the Farley complex for V . We then analyze the descending links of the polysimplicial complex, using a theorem of Belk and Forrest to prove increasing connectivity.
has introduced the higher-dimensional Thompson groups nV that are generalizations to the Thompson group V of self-homeomorphisms of the Cantor set and found a finite set of generators and relations in the case n = 2. We show how to generalize his construction to obtain a finite presentation for every positive integer n. As a corollary, we obtain another proof that the groups nV are simple (first proved by Brin).
Abstract. We find a lower bound to the size of finite groups detecting a given word in the free group. More precisely we construct a word w n of length n in non-abelian free groups with the property that w n is the identity on all finite quotients of size ∼ n 2/3 or less. This improves on a previous result of BouRabee and McReynolds quantifying the lower bound of the residual finiteness of free groups.
We describe the relation between two characterizations of conjugacy in groups of piecewise-linear homeomorphisms, discovered by Brin and Squier in [2] and Kassabov and Matucci in [5]. Thanks to the interplay between the techniques, we produce a simplified point of view of conjugacy that allows ua to easily recover centralizers and lends itself to generalization.
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 .
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