Groups, Geometry, and Dynamics

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Volume 6, Issue 2, 2012, pp. 279–315
DOI: 10.4171/GGD/158

The simultaneous conjugacy problem in groups of piecewise linear functions

Martin Kassabov (1) and Francesco Matucci (2)

(1) Department of Mathematics, Cornell University, 310 Malott Hall, NY 14853-4201, ITHACA, UNITED STATES
(2) Département de Mathématiques, Université Paris-Sud, Bâtiment 425, 91405, ORSAY CEDEX, FRANCE

Guba and Sapir asked if the simultaneous conjugacy problem is 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 orientation-preserving 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.

Keywords: Conjugacy problem, Thompson’s groups