This is one of the research seminars for the RAS program, that distinguishes itself from the postdocs and program associates seminars in that speakers are chosen among Research Members, Research Professors with occasional outside speakers.
Abstract: A finitely generated group G is profinitely rigid (in the absolute sense) if any finitely generated, residually finite group that has the same list of finite quotients as G must actually be isomorphic to G. The first examples of full-sized groups that are rigid in this sense were constructed in 2017 -- they are fundamental groups of 3-dimensional hyperbolic orbifolds with particular arithmetic properties. In this talk I shall explain how ideas from that construction can be extended to prove that certain cocompact Fuchsian groups are profinitely rigid (joint work with Reid, McReynolds and Spitler). I shall begin by sketching the history of profinite rigidity, with key examples, emphasising the connections with hyperbolic geometry (in both the classical and Gromovian senses). I shall end by outlining some of the ongoing work in the field.