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Talk

Geometry of Hyperconvex representations

  • James Farre (MPI MiS, Leipzig)
E2 10 (Leon-Lichtenstein)

Abstract

A discrete group of conformal automorphisms of the Riemann sphere is called a Kleinian group. Anosov Kleinian surface groups are quasi-conformal deformations of Fuchsian surface groups; they preserve a Jordan curve cutting the sphere into two disks on which the action of the group is nice. Bers proved that quasi-Fuchsian surface group representations are determined uniquely by the pair of conformal structures at infinity, giving a natural parameterization by a product of Teichmüller spaces. We study the higher rank analogue: hyperconvex Anosov surface group representations into PSL(d,C), introduced by Pozzetti—Sambarino—Wienhard. We define a natural map into a product of Teichmüller spaces of Riemann surface foliations and prove the following analogue of a famous theorem of Bowen from the 70’s: The Hausdorff dimension of the limit set of a fully hyperconvex surface subgroup into PSL(d,C) is equal to 1 iff it is conjugated into PSL(d,R). This is joint work with Beatrice Pozzetti and Gabriele Viaggi.

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