Groupes de Schottky et comptage

Quint, Jean-François

doi:10.5802/aif.2102

Cited by 23 publications

(22 citation statements)

References 11 publications

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“…These are finitely generated free subgroups of G, for which one has a good control on the relative position of the fixed points on F of the free generators, together with nice contraction properties. This precise information allows Quint [22] to build an equivariant continuous map, from the boundary at infinity of the group into F . The limit set is hence identified with a subshift of finite type.…”

Section: Introductionmentioning

confidence: 99%

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The orbital counting problem for hyperconvex representations

Sambarino¹

2015

Annales De l'Institut Fourier

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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 PSL(2, R)×PSL(2, R), induced by choosing two points on the Teichmüller space of the surface; and representations on the Hitchin component of PSL (d, R). We also prove a mixing property for the Weyl chamber flow in this setting.Theorem A (See Section 5). Let ρ : Γ → G be a Zariski-dense hyperconvex representation. Then there exist h, c > 0, and a probability measure µ on X F , such thatConsidering the constant function equal to 1, one obtains the following corollary.Corollary. Let ρ : Γ → G be a Zariski-dense hyperconvex representation. Then there exist h, c > 0, such thatThe exponential growth rate h in Theorem A is explicit: it is the topological entropy of a natural flow we construct, associated to the representation ρ. On the contrary, not much information is known about the constant c.As first shown by Margulis [18] in negative curvature, in order to obtain a counting theorem one usually proves a mixing property of a well chosen dynamical system. In compact manifolds with negative curvature, the geodesic flow plays this role. In infinite covolume, for example for convex cocompact groups, one should restrict the geodesic flow to its nonwandering set. When ∆ is a lattice in higher rank, use the mixing property of the Weyl chamber flow, to prove the counting result previously mentioned.Let τ be the Cartan involution on g = Lie(G), whose fixed point set is the Lie algebra of K. Consider p = {v ∈ g : τ v = −v} and a, a maximal abelian subspace contained in p. Denote by a + a closed Weyl chamber, and M the centralizer of exp(a) on K. The Weyl chamber flow is the right action by translations of exp(a) in ∆\G/M.

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Section: Introductionmentioning

confidence: 99%

“…The limit set is hence identified with a subshift of finite type. Quint [22] uses the Thermodynamic Formalism on this subshift, to obtain an exponential equivalence for the orbital counting problem.…”

Section: Introductionmentioning

confidence: 99%

The orbital counting problem for hyperconvex representations

Sambarino¹

2015

Annales De l'Institut Fourier

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“…Pour montrer cette loi des angles, on combinera astucieusement un théorème local limite pour les écarts modérés, une loi du logarithme itéré et une estimation de grandes déviations que nous avons démontrés dans ce but dans [4] en nous appuyant sur des idées de [11,18,13,5,1,2,7,23,14,12]. Ces arguments permettent aussi de décrire la tribu queue Q θ ∞ intersection des Q θ n , n ∈ N.…”

Section: Loi Des Anglesunclassified