THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS

Martin Bridgeman (1) , Richard Canary (2) , François Labourie (3) , Andres Sambarino (4)

1 BC - Boston College
2 Department of Mathematics - University of Michigan
3 UNS - Université Nice Sophia Antipolis (1965 - 2019)
4 UP11 - Université Paris-Sud - Paris 11

Using the thermodynamic formalism, we introduce a notion of intersection for projective Anosov representations, show analyticity results for the intersection and the entropy, and rigidity results for the intersection. We use the renormalized intersection to produce an Out(Γ)-invariant Riemannian metric on the smooth points of the deformation space of irreducible, generic, projective Anosov representations of a word hyperbolic group Γ into SLm(R). In particular, we produce mapping class group invariant Riemannian metrics on Hitchin components which restrict to the Weil–Petersson metric on the Fuchsian loci. Moreover, we produce Out(Γ)-invariant metrics on deformation spaces of convex cocompact representations into PSL 2 (C) and show that the Hausdorff dimension of the limit set varies analytically over analytic families of convex cocompact representations into any rank 1 semi-simple Lie group.

Domaines: Géométrie différentielle [math.DG]

Martin Bridgeman, Richard Canary, François Labourie, Andres Sambarino. THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS. Geometry and Functional Analysis, 2015, ⟨10.1007/s00039-015-0333-8⟩. ⟨hal-01141622⟩