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The geometry of quasi-Hitchin symplectic Anosov representations

Banff International Research Station for Mathematical Innovation and Discovery

In this talk we will discuss quasi-Hitchin representations in $\mathrm{Sp}(4,\mathbb{C})$, which are deformations of Fuchsian (and Hitchin) representations which remain Anosov. These representations acts on the space $\mathrm{Lag}(\mathbb{C}^4)$ of complex lagrangian grassmanian subspaces of $\mathbb{C}^4$. This theory generalises the classical and important theory of quasi-Fuchsian representations and their action on the Riemann sphere $\mathbb{C} P^1 = \mathrm{Lag} (\mathbb{C}^2)$. In the talk, after reviewing the classical theory, we will define Anosov and quasi-Hitchin representations and we will discuss their geometry. In particular, we show that the quotient of the domain of discontinuity for this action is a fiber bundle over the surface and we will describe the fiber. The projection map comes from an interesting parametrization of $\mathrm{Lag}(\mathbb{C}^4)$ as the space of regular ideal hyperbolic tetrahedra and their degenerations. (This is joint work with D.Alessandrini and A.Wienhard.)

Mathematics; Group theory and generalizations; Geometry; Algebraic structures

video/mp4

Author affiliation: University of Virginia

2020-06-07

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/74663

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/