arXiv research

The paper proves CMC foliations for quasi-Fuchsian manifolds near the Fuchsian locus.

An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient hyperbolic manifold contains a closed incompressible minimal surface with principal curvatures contained in (-1,1). We show that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius in the visual…

New foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus are uniquely determined.

Study foliations at infinity and constant mean curvature surfaces in quasi-Fuchsian manifolds.

The study classifies weakly almost Fuchsian manifolds and proves geometric properties.

Surveying Hitchin representations of Fuchsian groups.

We show that if a homeomorphism between the ideal boundaries of two Fuchsian buildings preserves the combinatorial cross ratio almost everywhere, then it extends to an isomorphism between the Fuchsian buildings. It follows that Mostow rigidity holds for Fuchsian buildings: if a group acts properly and cocompactly on tw…

Limit sets of $\mathrm{AdS}$-quasi-Fuchsian groups of $\mathrm{PO}(n,2)$ are always Lipschitz submanifolds. The aim of this article is to show that they are never $\mathcal{C}^1$, except for the case of Fuchsian groups. As a byproduct we show that $\mathrm{AdS}$-quasi-Fuchsian groups that are not Fuchsian are Zariski d…

Measuring foliations at infinity of quasi-Fuchsian manifolds, proving filling pairs and showing realisation.

We rigorously define the Liouville action functional for finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that the classical action - the critical point of the Liouville action functional, considered as a funct…

We prove that the renormalized volume of almost-Fuchsian hyperbolic $3$-manifolds is non-negative, with equality only for Fuchsian manifolds.

We compare critical exponent for quasi-Fuchsian groups acting on the hyperbolic 3-space, $\mathbb{H}^3$, and on invariant disks embedded in $\mathbb{H}^3$. We give a rigidity theorem for all embedded surfaces when the action is Fuchsian and a rigidity theorem for negatively curved surfaces when the action is quasi-Fuch…

Unique compact Fuchsian manifolds with convex boundary are determined by their boundary.

We study the Liouville action for quasi-Fuchsian groups with parabolic and elliptic elements. In particular, when the group is Fuchsian, the contribution of elliptic elements to the classical Liouville action is derived in terms of the Bloch-Wigner functions. We prove the first and second variation formulas for the cla…

The paper confirms a conjecture about foliating almost Fuchsian manifolds with CMC surfaces.

Study on Cheeger constant in specific hyperbolic 3-manifolds.

Let Gamma be a cocompact lattice in SO(1,n). A representation rho: Gamma \to SO(2,n) is quasi-Fuchsian if it is faithfull, discrete, and preserves an acausal subset in the boundary of anti-de Sitter space - a particular case is the case of Fuchsian representations, ie. composition of the inclusions of Gamma in SO(1,n) …

Random walks on Fuchsian Schottky groups have harmonic measures with lower dimension.

Study $\mathbb{C}$-Fuchsian subgroups of non-arithmetic lattices.

Alternative construction of quasi-Fuchsian flows using vortex equations.

We use Series' Markovian coding for words in Fuchsian groups and the Bowen-Series coding of limit sets to prove an ergodic theorem for Cesaro averages of spherical averages in a Fuchsian group.

The paper shows nearly-Fuchsian properties for certain hyperbolic 3-manifolds.

The trace set of a Fuchsian group $Γ$ ist the set of length of closed geodesics in the surface $Γ\backslash \mathbb{H}$. Luo and Sarnak showed that the trace set of a cofinite arithmetic Fuchsian group satisfies the bounded clustering property. Sarnak then conjectured that the B-C property actually characterizes arithm…

New insights into surface group actions and entropy.

We prove that any nonabelian, non-Fuchsian representation of a surface group into PSL(2,R) is the holonomy of a folded hyperbolic structure on the surface. Using similar ideas, we establish that any non-Fuchsian representation rho of a surface group into PSL(2,R) is strictly dominated by some Fuchsian representation j,…

We define a fuchsian affine action of a surface group to be such that the linear part factors through a representation of $SL(2,{\mathbb R})$. We prove a fuchsian affine action of a surface group is never proper.

The article constructs Fuchsian Schottky groups with conformal boundaries.

We prove the Gromov-Lawson-Rosenberg conjecture for cocompact Fuchsian groups, thereby giving necessary and sufficient conditions for a closed spin manifold of dimension greater than four with fundamental group cocompact Fuchsian to admit a metric of positive scalar curvature.

The hitting measure is singular and has dimension less than 1 for cocompact Fuchsian groups.

This paper shows how to approximate CAT(-1) representations by Fuchsian ones.

In this paper we show some properties of triangle invariants and shearing invariants of PSL(n,R)-Fuchsian representations. Moreover, using the Bonahon-Dreyer parameterization, we show that the Fuchsian locus of Hitchin components corresponds to a slice.

We consider the space of all quasifuchsian metrics on the product of a surface with the real line. We show that, in a neighborhood of the submanifold consisting of fuchsian metrics, every non-fuchsian metric is completely determined by the bending data of its convex core.

New proof shows Fuchsian groups have irrational length spectra.

In the study of Fuchsian groups, it is a nontrivial problem to determine a set of generators. Using a dynamical approach we construct for any cocompact arithmetic Fuchsian group a fundamental region in $\mathbf{SL}_2(\mathbb{R})$ from which we determine a set of small generators.

In this paper we give the characterization of Fuchsian groups acting on quaternionic hyperbolic 2-space.

In this paper, we study Fuchsian loci of ${\rm PSL}_n(\mathbb{R})$-Hitchin components. In particular, using the Bonahon-Dreyer parametrization of ${\rm PSL}_n(\mathbb{R})$-Hitchin components, we give an explicit parametrization of Fuchsian loci of a pair of pants.

The paper characterizes geometrically finite surfaces via geodesic covers.

We show that every hyperbolic knot complement contains a closed quasi-Fuchsian surface.

New representations preserve hyperbolicity but not Fuchsian property.

The hyperbolic space $ \H^d$ can be defined as a pseudo-sphere in the $(d+1)$ Minkowski space-time. In this paper, a Fuchsian group $Γ$ is a group of linear isometries of the Minkowski space such that $\H^d/Γ$ is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, g…

Study compares metrics from negative curvature and quasi-Fuchsian representations.

A new metric model for quasi-Fuchsian space defined by Bers metrics.

Holomorphic connections found on Riemann surfaces with Fuchsian monodromy.

In this paper, we study the topology of the boundaries of quasi-Fuchsian spaces. We first show for a given convergent sequence of quasi-Fuchsian groups, how we can know the end invariant of the limit group from the information on the behaviour of conformal structures at infinity of the groups. This result gives rise to…

Formula derived for FUP exponent in quasi-Fuchsian groups.

Study on non-classical generating sets in Fuchsian Schottky groups.

A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric to a hyperbolic…

The aim of this paper is to relate Thurston's metric on Teichmüller space to several ideas initiated by T. Sorvali on isomorphisms between Fuchsian groups. In particular, this will give a new formula for Thurston's asymmetric metric for surfaces with punctures. We also update some results of Sorvali on boundary isomorp…