Infinitesimal rigidity is established for irreducible cyclic surfaces and n-alternating surfaces in H^{p,q}, unifying prior results on maximal space-like surfaces, alternating holomorphic curves, and A-surfaces.
Maximal Representations of Surface Groups: Symplectic Anosov Structures
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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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A comprehensive introduction to spectral networks that develops higher-rank Teichmüller theory in parallel with class S gauge theory and BPS spectra.
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Infinitesimal Rigidity of Cyclic Surfaces and Alternating Surfaces
Infinitesimal rigidity is established for irreducible cyclic surfaces and n-alternating surfaces in H^{p,q}, unifying prior results on maximal space-like surfaces, alternating holomorphic curves, and A-surfaces.
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Spectral Networks: Bridging higher-rank Teichm\"uller theory and BPS states
A comprehensive introduction to spectral networks that develops higher-rank Teichmüller theory in parallel with class S gauge theory and BPS spectra.