In this talk we give a brief introduction to Thompson's groups F, T and V, three subgroups of the group of homeomorphisms of the Cantor set. They were defined by Richard Thompson in 1965, who used them to construct finitely presented groups with unsolvable word problems. We then study the solution of Belk-Matucci to the conjugacy problem for these groups, using a topological approach.

 

Next we will introduce a larger class of subgroups of the homeomorphism group of the Cantor set, a family of Thompson-like groups. We will explain how to extend the ideas of Belk-Matucci to this setting, in order to solve the conjugacy problem, In addition, this implies that certain pairs of these Thompson-like groups are non-isomorphic.

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