The orbital counting problem for hyperconvex representations

By Andr'es Sambarino. 2012

66 citations recorded by the source corpus.

We give a precise counting result on the symmetric space of a noncompact real algebraic semisimple group $G,$ for a class of discrete subgroups of $G$ that contains, for example, representations of a surface group on $\textrm{PSL}(2,\mathbb R)\times\textrm{PSL}(2,\mathbb R),$ induced by choosing two points on the Teichm\"uller space of the surface; and representations on the Hitchin component of $\textrm{PSL}(d,\mathbb R).$ We also prove a mixing property for the Weyl chamber flow in this setting.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c08843c2015cea0dfac34fc48be5d75185487b94

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

AN INVITATION TO HIGHER TEICHMÜLLER THEORY (2019, unknown): “Peter Sarnak (2014).”
Anosov representations and dominated splittings (2016, background): “ons of G, moreover, it keeps track of Gromov products for all irreducible representations of G. Indeed one has the following consequence of equation (8.3). 26This is the opposite of the one dened in [Sa]. ANOSOV REPRESENTATIONS AND DOMINATED SPLITTINGS 51 Remark 8.11. Let : GÑPSLpVqbe a nite dimensi”
Eigenvalues and entropy of a Hitchin representation (2014, background): “Labourie [24] shows that a representation in Hitchin(Σ, d) (from now on a Hitchin representation) is discrete, irreducible and faithful, and consists of purely loxodromic elements.”
Anosov groups: local mixing, counting and equidistribution (2020, methodology): “Using the reparametrization theorem of [4] for a general case, Chow and Sarkar [8] proved the following theorem; their proof uses a different symbolic coding than [45] and works for functions which are not necessarily M –invariant.”
An invitation to higher Teichm\"uller theory (2018, background): “Other important quantities that have been investigated using this dynamical viewpoint are the critical exponent and the topological entropy of Hitchin representations, which are related to counting orbit points on the symmetric space [99, 98, 94].”
Invariant Measures for Horospherical Actions and Anosov Groups (2020, background): “In view of this lemma, our definition of Gromov product differs by − i from the one given in [36].”