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Maximal Representations of Surface Groups: Symplectic Anosov Structures

Marc Burger·Alessandra Iozzi·Francois Labourie·Anna Wienhard
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Abstract

Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Gamma be the fundamental group of a compact orientable surface of genus at least 2. We survey the study of maximal representations, that is the subset of Hom(Gamma,G) which is a union of components characterized by the maximality of the Toledo invariant. Then we concentrate on the particular case G=SP(2n,R), and we show that the image of Gamma under any maximal representation is a discrete faithful realization of Gamma as a Kleinian group of complex motions in X with an associated Anosov system, and whose limit set in an appropriate compactification of X is a rectifiable circle.

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Marc Burger, Alessandra Iozzi, Francois Labourie, Anna Wienhard. 2005-10-07. Maximal Representations of Surface Groups: Symplectic Anosov Structures. https://arxiv.org/abs/math/0506079

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