Seminar

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The work of Bridgeman-Canary-Labourie-Sambarino associates a metric Anosov flow to any Anosov representation. This has led to prove several counting and rigidity results, as well as structural results for the deformation spaces of these representations. For instance, the space of reparameterizations of the geodesic flow of a hyperbolic surface contains all its Hitchin components, so it can be thought of as a "highest Teichmüller space".

In this talk, I'll explain a correspondence between reparameterizations of the (Mineyev) flow space of a hyperbolic group and geometric actions of the group on Gromov hyperbolic spaces. In this setting, we also prove continuity of the Bowen-Margulis-Sullivan geodesic current map, related to the measure of maximal entropy for the associated reparameterization. In particular, the BMS map induces an embedding of Hitchin components into the space of projective geodesic currents, which in higher rank is different from the embedding coming from the positively ratioed perspective.

This is joint work with Stephen Cantrell and Dídac Martínez-Granado.

A geometric correspondence for reparameterizations of geodesic flows