Conjugacy in Thompson’s groups

We give a unified solution the conjugacy problem in Thompson’s groups F , V , and T using strand diagrams, a modification of tree diagrams. We then analyze the complexity of the resulting algorithms. Thompson’s group F is the group of all piecewise-linear homeomorphisms of the unit interval satisfying the following conditions: 1. Every slope is a power of two, and 2. Every breakpoint has dyadic rational coordinates. The group F is finitely presented (with two generators and two relations) and torsion-free. In addition to F , Thompson introduced two other finitely-generated groups known as T and V . Briefly, T is the set of piecewise-linear self-homeomorphisms of the circle [0, 1] / {0, 1} satsifying the two conditions above, while V is the set of piecewise-linear bijections of the interval (or self-homeomorphisms of the Cantor set) satisfying the above conditions. The standard introduction to F , T , and V is [8]. We will assume some familiarity with Thompson’s groups and, in particular, with tree diagrams. In this paper we give a unified solution for the conjugacy problem in Thompson’s groups F, T and V . We introduce strand diagrams, a modification of tree diagrams for these groups, and show how identifying the roots of the two trees defines a conjugacy invariant in all cases. This reduces the conjugacy problem to the study of the isomorphism problem for certain classes of graphs and gives us elementary proofs of some known results. Strand diagrams were first introduced by Pride in his study of the homotopy of relations using the term pictures in [19], [20] and [3] and are dual to the diagrams introduced by Guba and Sapir in [11]. In 1997 Guba and Sapir showed that F can be viewed as a diagram group for the monoid presentation 〈x | x = x〉 [11]. They give a solution for each diagram The first author gratefully acknowledges partial support from an NSF Postdoctoral Research Fellowship while he was at Texas A&M University. This work is part of the second author’s PhD thesis at Cornell University. The second author gratefully acknowledges the Centre de Recerca Matemàtica (CRM) and its staff for the support received during the completion of this work.