Speaker
Description
Thompson’s group $ V $ was the first known finitely presented infinite simple group. The ordinary conjugacy problem in $ V $ was first solved by Higman and later given a uniform combinatorial treatment, via reduced closed abstract strand diagrams, by Belk–Matucci. The simultaneous (or $ k $-simultaneous) conjugacy problem asks whether, given two $ k $-tuples $ (g_1,\dots,g_k) $ and $ (h_1,\dots,h_k) $ of elements of a group $ G $, there exists a single conjugator $ c\in G $ such that $ c^{-1}g_ic=h_i $ for every $ i $. This problem had remained open for $ V $.
In this talk we solve the simultaneous conjugacy problem in $ V $. We exhibit a decision procedure that, given any two $ k $-tuples, determines whether a common conjugator exists and, when one does, produces such an element. The method extends previously existing algorithms and uses the results of Bleak-Bowman-Gordon-Graham-Hughes-Matucci-Sapir on centralisers together with a careful analysis of the possible ways their dynamical data can be simultaneously matched under a single conjugating transformation. Joint work with Gemma Crowe.