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

Abstract: Let \(F\) be a finitely generated free group. Let \(w_1\) and \(w_2\) be suitably random/generic elements in \(F\). Consider the HNN extension \( G = \langle F, t \,{\mid}\, t w_1 t^{-1} = w_2 \rangle\). It is already known that \(G\) will be one-ended and hyperbolic. What we have shown is that \(G\) is quasi-isometrically rigid. That is, if a finitely generated group \(H\) is quasi-isometric to \(G\), then \(G\) and \(H\) are virtually isomorphic. The main argument involves applying a new proof of Leighton's graph covering theorem.

Our full result is for finite graphs of groups with virtually free vertex groups and and two-ended edge groups. However the statement here is more technical; in particular, not all such groups are quasi-isometrically rigid.

This is joint work with Sam Shepherd.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

( slides )

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Geometry and topology online

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