Cataclysm deformations for Anosov representations

Abstract

In this thesis, we construct special deformations for Anosov representations, so-called cataclysms, and investigate their properties. In Teichmuller theory, cataclysms and the closely related shearing coordinates carry information about the structure of Teichmuller space. Therefore, the question arises if cataclysms also exist in higher Teichmuller spaces or more generally for Anosov representations. We construct cataclysms for θ-Anosov representations into a semisimple non-compact connected real Lie group G, where θ ⊂ Δ is a subset of the simple roots that is invariant under the opposition involution. Important steps in our construction of cataclysm deformations are the definition of the appropriate parameter space as well as the definition of slithering maps in the context of θ-Anosov representations. These maps generalize slithering maps for Hitchin representations which were defined by Bonahon and Dreyer in their parametrization of the Hitchin component. We then construct stretching maps, shearing maps and finally cataclysms. Cataclysms have some natural properties: They are additive and behave well with respect to composing an Anosov representation with a Lie group homomorphism. Moreover, we show how the cataclysm deformation of an Anosov representation affects the associated boundary map. In Teichmuller space, cataclysm deformations are injective. However, this does not hold true for θ-Anosov representations in general. We give sufficient conditions for injectivity as well as for non-injectivity of the deformation. For certain classes of reducible representations, we explicitly determine the subspace of the parameter space on which the deformation is trivial. These representations include a family of Borel Anosov representations into SL(2n+1,R), by which we show that cataclysms of Borel Anosov representations are not necessarily injective.