Academic literature on the topic 'Convex-Cocompact representations'

AbstractLet (M,gλ) be a 𝒞2-family of complete convex-cocompact metrics with pinched negative sectional curvatures on a fixed manifold. We show that the topological entropy htop(gλ) of the geodesic flow is a 𝒞1 function of λ and we give an explicit formula for its derivative. We apply this to show that if ρλ(Γ)⊂PSL2(ℂ) is an analytic family of convex-cocompact faithful representations of a Kleinian group Γ, then the Hausdorff dimension of the limit set Λρλ(Γ) is a 𝒞1 function of λ. Finally, we give a variation formula for Λρλ (Γ).

In this paper, we study an interesting curve, the so-called Manhattan curve, associated with a pair of boundary-preserving Fuchsian representations of a (non-compact) surface; in particular, representations corresponding to Riemann surfaces with cusps. Using thermodynamic formalism (for countable state Markov shifts), we prove the analyticity of the Manhattan curve. Moreover, we derive several dynamical and geometric rigidity results, which generalize results of Burger [Intersection, the Manhattan curve, and Patterson–Sullivan theory in rank 2. Int. Math. Res. Not.1993(7) (1993), 217–225] and Sharp [The Manhattan curve and the correlation of length spectra on hyperbolic surfaces. Math. Z.228(4) (1998), 745–750] for convex cocompact Fuchsian representations.

Abstract For any integer d ≥ 1 {d\geq 1} , we obtain counting and equidistribution results for tori with small volume for a class of d-dimensional torus packings, invariant under a self-joining Γ ρ < ∏ i = 1 d PSL 2 ⁡ ( ℂ ) {\Gamma_{\rho}<\prod_{i=1}^{d}\operatorname{PSL}_{2}(\mathbb{C})} of a Kleinian group Γ formed by a d-tuple of convex-cocompact representations ρ = ( ρ 1 , … , ρ d ) {\rho=(\rho_{1},\dots,\rho_{d})} . More precisely, if 𝒫 {\mathcal{P}} is a Γ ρ {\Gamma_{\rho}} -admissible d-dimensional torus packing, then for any bounded subset E ⊂ ℂ d {E\subset\mathbb{C}^{d}} with ∂ ⁡ E {\partial E} contained in a proper real algebraic subvariety, we have lim s → 0 ⁡ s δ L 1 ⁢ ( ρ ) ⋅ # ⁢ { T ∈ 𝒫 : Vol ⁡ ( T ) > s , T ∩ E ≠ ∅ } = c 𝒫 ⋅ ω ρ ⁢ ( E ∩ Λ ρ ) . \lim_{s\to 0}{s^{\delta_{L^{1}}({\rho})}}\cdot\#\{T\in\mathcal{P}:% \operatorname{Vol}(T)>s,\,T\cap E\neq\emptyset\}=c_{\mathcal{P}}\cdot\omega_{% \rho}(E\cap\Lambda_{\rho}). Here δ L 1 ⁢ ( ρ ) {\delta_{L^{1}}(\rho)} , 0 < δ L 1 ⁢ ( ρ ) ≤ 2 / d {0<\delta_{L^{1}}(\rho)\leq 2/\!{\sqrt{d}}} , denotes the critical exponent of the self-joining Γ ρ {\Gamma_{\rho}} with respect to the L 1 {L^{1}} -metric on the product ∏ i = 1 d ℍ 3 {\prod_{i=1}^{d}\mathbb{H}^{3}} , Λ ρ ⊂ ( ℂ ∪ { ∞ } ) d {\Lambda_{\rho}\subset(\mathbb{C}\cup\{\infty\})^{d}} is the limit set of Γ ρ {\Gamma_{\rho}} , and ω ρ {\omega_{\rho}} is a locally finite Borel measure on ℂ d ∩ Λ ρ {\mathbb{C}^{d}\cap\Lambda_{\rho}} which can be explicitly described. The class of admissible torus packings we consider arises naturally from the Teichmüller theory of Kleinian groups. Our work extends previous results of [H. Oh and N. Shah, The asymptotic distribution of circles in the orbits of Kleinian groups, Invent. Math. 187 2012, 1, 1–35] on circle packings (i.e., one-dimensional torus packings) to d-torus packings.

L'objectif de cette thèse est d'étudier et de construire des groupes discrets agissant par isométries sur l'espace hyperbolique de dimension infinie. L'espace hyperbolique de dimension infinie est une variété riemannienne symétrique de dimension infinie et de courbure constante égale à -1. Son étude (ainsi que celle d'autres espaces symétriques de type non compact et de dimension infinie) a été suggérée par Gromov dans l'ouvrage intitulé "Asymptotic invariants of infinite groups". Il y souligne en particulier la nécessité de définir la notion de "groupes discrets" dans ce contexte. Les espaces hyperboliques de dimension finie et leurs groupes d'isométries discrets ont été très largement étudiés pour leurs liens avec les variétés hyperboliques. Une propriété bien établie dans ce domaine est la stabilité des représentations convexes cocompactes dans les groupes d'isométries d'espaces hyperboliques en dimension finie. À partir d'une observation de Monod et Py, nous établissons un résultat similaire pour des représentations en dimension infinie. Ce résultat de stabilité suggère que l'on peut ensuite déformer des représentations convexes cocompactes de groupes de type fini. De telles représentations existent grâce à une classification de Monod et Py et l'on montre que pour un groupe de surface, l'espace des déformations d'une représentation convexe cocompacte (en dimension infinie) est de dimension infinie. Tous les groupes obtenus par déformation sont des groupes d'isométries "fortement discrets" de l'espace hyperbolique de dimension infinie. Pour trouver d'autres exemples de groupes discrets agissant sur les espaces hyperboliques, on peut se tourner vers les groupes de Coxeter. Leurs actions par réflexions peuvent être décrites très simplement à l'aide d'une matrice. Ce sont donc de bons candidats pour construire des groupes discrets en dimension infinie. Nous donnons une condition suffisante, inspirée par la théorie de Vinberg, pour que des groupes de Coxeter de type infini agissent de manière irréductible sur l'espace hyperbolique de dimension infinie et décrivons quelques exemples de groupes satisfaisant ce critère. Cependant, les propriétés de discrétude ne semblent pas se généraliser aux groupes de type infini
This thesis aims at studying and constructing discrete groups acting by isometries on the infinite-dimensional hyperbolic space. The infinite-dimensional hyperbolic space is a Riemannian symmetric space of infinite dimension and constant curvature equal to -1. Its study (and that of other symmetric spaces of non-compact type and infinite dimension) was suggested by Gromov in its work entitled "Asymptotic invariants of infinite groups". In particular, he emphasises the need to define the notion of "discrete groups" in this context. Finite-dimensional hyperbolic spaces and their discrete groups of isometries have been largely studied for their relations with hyperbolic manifolds. A well-established property in this field is the stability of convex-cocompact representations into the isometry groups of finite-dimensional hyperbolic spaces. From an observation by Monod and Py, we prove a similar statement for infinite-dimensional representations. This stability result suggests that one can deform convex cocompact representations of finitely generated groups. Such representations do exist thanks to a classification by Monod and Py and we show that for a surface group, the space of deformations of convex-cocompact (infinite-dimensional) representations has infinite dimension. All the groups obtained by deformations are "strongly discrete" groups of isometries of the infinite-dimensional hyperbolic space. To find other examples of discrete groups acting on hyperbolic spaces, one can think of Coxeter groups. They admit actions by reflections that can be easily described using some matrix. Thus, they are interesting candidates to provide discrete groups in infinite dimension. Inspired by Vinberg's theory, we give a sufficient condition for infinitely generated Coxeter groups to act irreducibly on the infinite-dimensional hyperbolic space and we discuss some examples of groups satisfying our criterion. However, it seems that the discreteness properties do not pass to infinitely generated groups