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Quasi-isometric rigidity of graphs of free groups with cyclic edge groups

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Let F be a finitely rank free group. Let w_1 and w_2 be suitable random/generic elements in F. Consider the HNN extension G generated by F and a stable letter t, with relation t w_1 t^{-1} = w_2 . It is known from existing results that G will be 1-ended and hyperbolic. We have shown that G is quasi-isometrically rigid. That is to say that if a f.g. group H is quasi-isometric to G, then G and H are virtually isomorphic. The full result is for finite graphs of groups with virtually free vertex groups and two-ended edge groups, but the statement is more technical—not all such groups are QI-rigid. The main argument involves applying a new proof of Leighton’s graph covering theorem.

This is joint work with Sam Shepherd.

This talk is part of the Geometric Group Theory (GGT) Seminar series.

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