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Giuseppe Martone, PMA 12.166: The Hilbert pressure metric for the space of finite area convex projective surfaces
Monday, May 02, 2022, 02:00pm - 03:00pm
A finite area real convex projective surface is the quotient of a strictly convex domain in the projective plane by a free and properly discontinuous action of a discrete subgroup of SL(3,R). For the purposes of this talk, the surface has negative Euler characteristic and, possibly, punctures. A finite area real convex projective surface is naturally equipped with a length function for closed curves, called the Hilbert length. Finite area hyperbolic surfaces and their hyperbolic length function provide a large family of examples. The holonomies of finite area real convex projective structures on a given topological surface S single out a subspace H(3,S) of the character variety of the fundamental group of S into the Lie group SL(3,R). The space H(3,S) is topologically trivial and it is a prominent example of a higher Teichmuller space. In this talk, we describe the construction of a mapping class group invariant path-metric on H(3,S) associated to the Hilbert length and explicitly describe its degeneracy. This construction, which is new even when S is a closed surface, is motivated by McMullen's dynamical interpretation of the Weil-Petersson metric on Teichmuller space and by Bridgeman, Canary, Labourie, Sambarino's pressure metrics. The key dynamical ingredients in our construction come from the Thermodynamic Formalism of countable Markov shifts. This talk is based on joint work with Harry Bray, Dick Canary and Nyima Kao.
Location: PMA 12.166

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