Topological Invariants of Anosov Representations

We define new topological invariants for Anosov representations and study them in detail for maximal representations of the fundamental group of a closed oriented surface into the symplectic group.Comment: 66 pages, several changes, some consequences adde

Higher Teichm\"uller Spaces: from SL(2,R) to other Lie groups

The first part of this paper surveys several characterizations of Teichm\"uller space as a subset of the space of representation of the fundamental group of a surface into PSL(2,R). Special emphasis is put on (bounded) cohomological invariants which generalize when PSL(2,R) is replaced by a Lie group of Hermitian type. The second part discusses underlying structures of the two families of higher Teichm\"uller spaces, namely the space of maximal representations for Lie groups of Hermitian type and the space of Hitchin representations or positive representations for split real simple Lie groups.Comment: The file uploaded on May 12th was the wrong one and did not contain the Section 4.6 that was added. This is the version to appear in the Handbook of Teichm\"uller theor

Limits of geometries

A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spaces, describing the basic process by which one homogeneous geometry may transform into another. We develop a general framework to describe transitions in the context that both geometries involved are represented as sub-geometries of a larger ambient geometry. Specializing to the setting of real projective geometry, we classify the geometric limits of any sub-geometry whose structure group is a symmetric subgroup of the projective general linear group. As an application, we classify all limits of three-dimensional hyperbolic geometry inside of projective geometry, finding Euclidean, Nil, and Sol geometry among the limits. We prove, however, that the other Thurston geometries, in particular $\mathbb{H}^2 \times \mathbb{R}$ and $\widetilde{\operatorname{SL}_2 \mathbb{R}}$, do not embed in any limit of hyperbolic geometry in this sense.Comment: 40 pages, 2 figures. new in v2: figure 2 added, minor edits to Sections 1,2,