Fanny Kassel

Anosov representations and proper actions

We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan projection or Lyapunov projection of the Lie group. Using a properness criterion of Benoist and Kobayashi, we derive applications to proper actions on homogeneous spaces of reductive groups.

Poincaré series for non-Riemannian locally symmetric spaces

Abstract We initiate the spectral analysis of pseudo-Riemannian locally symmetric spaces Γ \ G / H , beyond the classical cases where H is compact (automorphic forms) or Γ is trivial (analysis on symmetric spaces). For any non-Riemannian reductive symmetric space X = G / H on which the discrete spectrum of the Laplacian is nonempty, and for any discrete group of isometries Γ whose action on X is “sufficiently proper”, we construct L 2 -eigenfunctions of the Laplacian on X Γ : = Γ \ X for an infinite set of eigenvalues. These eigenfunctions are obtained as generalized Poincare series, i.e. as projections to X Γ of sums, over the Γ-orbits, of eigenfunctions of the Laplacian on X. We prove that the Poincare series we construct still converge, and define nonzero L 2 -functions, after any small deformation of Γ inside G, for a large class of groups Γ. Thus the infinite set of eigenvalues we construct is stable under small deformations. This contrasts with the classical setting where the nonzero discrete spectrum varies on the Teichmuller space of a compact Riemann surface. We actually construct joint L 2 -eigenfunctions for the whole commutative algebra of invariant differential operators on X Γ .

Compact anti-de Sitter 3-manifolds and folded hyperbolic structures on surfaces

We prove that any nonabelian, non-Fuchsian representation of a surface group into PSL(2,R) is the holonomy of a folded hyperbolic structure on the surface. Using similar ideas, we establish that any non-Fuchsian representation rho of a surface group into PSL(2,R) is strictly dominated by some Fuchsian representation j, in the sense that the hyperbolic translation lengths for j are uniformly larger than for rho; conversely, any Fuchsian representation j strictly dominates some non-Fuchsian representation rho, whose Euler class can be prescribed. This has applications to compact anti-de Sitter 3-manifolds.

Convex cocompactness in pseudo-Riemannian hyperbolic spaces

Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups

Margulis spacetimes via the arc complex

\mathrm {PO}(p,q)

Some open questions on anti-de Sitter geometry

PO(p,q) by considering their action on the associated pseudo-Riemannian hyperbolic space

Maximally stretched laminations on geometrically finite hyperbolic manifolds

\mathbb {H}^{p,q-1}