Hyperconvex representations and exponential growth

By Andr'es Sambarino. Ergodic Theory and Dynamical Systems, 2012

65 citations recorded by the source corpus.

Abstract Let $G$ be a real algebraic semi-simple Lie group and $\Gamma $ be the fundamental group of a closed negatively curved manifold. In this article we study the limit cone, introduced by Benoist [Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal. 7(1) (1997), 1–47], and the growth indicator function, introduced by Quint [Divergence exponentielle des sous-groupes discrets en rang supérieur. Comment. Math. Helv. 77 (2002), 503–608], for a class of representations $\rho : \Gamma \rightarrow G$ admitting an equivariant map from $\partial \Gamma $ to the Furstenberg boundary of the symmetric space of $G, $ together with a transversality condition. We then study how these objects vary with the representation.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e9929fa78c17becc105e3e48d00319d8de5cb1a1

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Citations in the corpus

Invariant Measures for Horospherical Actions and Anosov Groups (2020, background): “2) int(PLΓ) ' D Γ := {ψ ∈ a∗ : ψ ≥ ψΓ, ψ(v) = ψΓ(v) at some v ∈ intLΓ} where int(PLΓ) denotes the interior of PLΓ ([31], [37], [6], cf.”
An invitation to higher Teichm\"uller theory (2018, methodology): “A key aspect in the work of Potrie and Sambarino has been the regularity of the map ξ : S1 → G/B of a Hitchin representation.”
AN INVITATION TO HIGHER TEICHMÜLLER THEORY (2019, unknown): “Andrés Sambarino (2014b).”
Anosov groups: local mixing, counting and equidistribution (2020, methodology): “6 is the following mixing result for the Bowen-Margulis-Sullivan measures proved by Thirion [43] for Schottky groups and by Sambarino [38] for Anosov groups which arise from representations of the fundamental group of a closed negatively curved Riemannian manifold, using thermodynamic formalism.”
Eigenvalues and entropy of a Hitchin representation (2014, background): “The main purpose of this section and Section 4 is to extend several results from [34] and [33] to the Anosov representations setting.”
The pressure metric for Anosov representations (2013, unknown): “D G ] 2 1”
The orbital counting problem for hyperconvex representations (2012, background): “The point g θ − is called the repelling hyperplane of g. Lemma 4.2 ([26, Section 3]) .”
Conformality for a robust class of non-conformal attractors (2019, methodology): “roposition 9.3), and discuss a new proof and generalization of a result of Potrie-S. [43]. The two appendices of the paper rely on the Thermodynamical formalism: in Appendix A we extend results of S. [49] and Bridgeman-Canary-Labourie-S. [8] to show continuity of the entropy map on the character var”