Critical exponent and Hausdorff dimension in pseudo-Riemannian hyperbolic geometry

Glorieux, Olivier; Monclair, Daniel

doi:10.48550/arxiv.1606.05512

Cited by 3 publications

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“…This happens exactly when q restricts to a signature (1, 1) quadratic form on x ⊕ y. The following was proved in [GM16][Proposition 3.2] Lemma 4.1. Let x, y be two distinct points in H 2,n and x 0 , y 0 be vectors in x and y respectively satisfying q(x 0 ) = q(y 0 ) = −1.…”

Section: Maximal Surfaces In Pseudo-hyperbolic Spacesmentioning

confidence: 99%

Quasicircles and quasiperiodic surfaces in pseudo-hyperbolic spaces

Labourie¹,

Toulisse²

2020

Preprint

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We study in this paper quasiperiodic maximal surfaces in pseudohyperbolic spaces and show that they are characterised by a curvature condition, Gromov hyperbolicity or conformal hyperbolicity. We show that the limit curves of these surfaces in the Einstein Universe admits a canonical quasisymmetric parametrisation, while conversely every quasisymmetric curve in the Einstein Universe bounds a quasiperiodic surface in such a way that the quasisymmetric parametrisation is a continuous extension of the uniformisation; we give applications of these results to asymptotically hyperbolic surfaces, rigidity of Anosov representations and a version of the universal Teichmüller space.

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Section: Maximal Surfaces In Pseudo-hyperbolic Spacesmentioning

confidence: 99%

Quasicircles and quasiperiodic surfaces in pseudo-hyperbolic spaces

Labourie¹,

Toulisse²

2020

Preprint

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“…Those projectively convex cocompact groups preserving a non degenerate quadratic form are precisely the H p,q -convex cocompact groups introduced in [DGK18], whose critical exponent was studied by the first two authors in [GM18]. In this setting, Theorem 1.1 gives an alternative proof of the inequality…”

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confidence: 99%

Hausdorff dimension of limit sets for projective Anosov representations

Glorieux¹,

Monclair²,

Tholozan³

2019

Preprint

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“…For certain classes of Anosov subgroups, Patterson-Sullivan theory was used by Sambarino in [Sam14,Sam15] to solve certain counting problems, while in [BCLS15] Bridgemann-Canary-Labourie-Sambarino used related thermodynamic formalism to construct pressure metrics on spaces of Hitchin representations. Moreover, Glorieux-Monclair ( [GM16]) studied the Patterson-Sullivan theory in the case of convex-cocompact subgroups of the isometry group of H p,q equipped with the pseudo-Riemannian metric.…”

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confidence: 99%

Patterson-Sullivan theory for Anosov subgroups

Dey,

Kapovich

2019

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We extend several notions and results from the classical Patterson-Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit sets, computed with respect to a suitable Gromov (pre-)metric on the flag manifold, and the Finsler critical exponents of Anosov subgroups.Let G be a noncompact real semisimple Lie group, X = G/K be the associated symmetric space and Γ be a τ mod -Anosov subgroup of G. We will be assuming several conditions on G and X; they are labeled as "assumption" in Section 1. We consider two types of G-invariant (pseudo-)metrics on X, namely, one is the Riemannian metric of the symmetric space X and the other one is Finslerian. The critical exponents of Γ with respect to these two metrics, denoted by δ R and δ F , respectively, are defined in the usual fashion, i.e., as the exponents of convergence of associated Poincaré series (see Section 2). Using the classical construction of Patterson, we define a Γ-invariant conformal density on the flag limit set of Γ (see Section 3).Throughout this paper, the Finsler metric is given more emphasis than its Riemannian counterpart. For example, the construction of the above mentioned Patterson-Sullivan density is carried out in terms of the Finsler metric. The main reason for this choice is that Finsler metrics reflect the asymptotic geometry of Γ better than the Riemannian metric.We should note that many of the results in this paper are often proven for more general classes of discrete subgroups of G with the hope that the results may be useful, for instance, in the study of relatively Anosov subgroups.1 Regarding Anosov subgroups, the main results of this paper are summarized below.Let σ mod be a maximal simplex in the Tits building of X, ι : σ mod → σ mod be the opposition involution, τ mod be an ι-invariant face of σ mod , P be the maximal parabolic subgroup of G that stabilizes τ mod , and Flag(τ mod ) = G/P be the partial flag manifold associated to the face τ mod (see Subsection 1.2). Main theorem.Let Γ be a nonelementary τ mod -Anosov subgroup of G and δ F be the Finsler critical exponent for the action of Γ on the symmetric space X = G/K. Then the Patterson-Sullivan density µ (constructed with respect to the Finsler metric on X) on the flag limit set Λ τ mod (Γ) ⊂ Flag(τ mod ) is the unique (up to a constant factor) Γ-invariant conformal density. Moreover, (i) The density µ is non-atomic and its dimension equals to δ F .(ii) The support of µ is Λ τ mod (Γ) and the action Γ Λ τ mod (Γ) is ergodic with respect to µ.(iii) The critical exponent δ F (as well as the Riemannian critical exponent δ R ) is positive and finite.(iv) The Poincaré series of Γ diverges at the critical exponent δ F . In other words, Γ has (Finsler) divergence type.(v) The δ F -dimensional Hausdorff measure on Λ τ mod (Γ) with respect to a Gromov (pre-)metric2 is a member of a Γ-invariant confo...

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Critical exponent and Hausdorff dimension in pseudo-Riemannian hyperbolic geometry

Cited by 3 publications

References 0 publications

Quasicircles and quasiperiodic surfaces in pseudo-hyperbolic spaces

Quasicircles and quasiperiodic surfaces in pseudo-hyperbolic spaces

Hausdorff dimension of limit sets for projective Anosov representations

Patterson-Sullivan theory for Anosov subgroups

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