Maximal automorphisms found for complex projective structures.
problem Maximizing automorphisms in complex projective structures.
method Analyzing Fuchsian uniformizations and Galois Belyi curves.
result Fuchsian uniformizations of Hurwitz surfaces achieve maximal automorphisms.
The paper uses Tannakian reconstruction to understand hyperbolic log-orbi curves.
problem Understanding the structure of hyperbolic log-orbi curves.
method Formulates hyperbolic uniformization as a Tannakian reconstruction theorem and constructs a canonical maximal parahoric PSL2-Higgs object.
result Reconstructs the absolute Galois group of a one-variable complex function field as the inverse limit of etale fundamental groups of orbifold models.
Complex projective structures can be joined via simple bubbling and debubbling.
problem Joining complex projective structures with quasi-Fuchsian holonomy.
method Performing (de)grafting via a sequence of one bubbling and one debubbling.
result Any complex projective structure can be joined to the uniformizing structure by a simple sequence of one bubbling and one debubbling.
Geometric interpretation of Fock-Goncharov positivity and disk stabilization in symmetric space.
problem Understanding Fock-Goncharov positivity and its geometric implications.
method Geometric interpretation and bending deformations of Fuchsian representations.
result Stabilization of a uniform Finsler quasi-convex disk in the symmetric space.
A new metric model for quasi-Fuchsian space defined by Bers metrics.
problem Understanding the quasi-Fuchsian space of a surface.
method Introducing Bers metrics and studying their properties to model QF(S).
result New integral representations of the Goldman symplectic form and holomorphic extension of the Weil-Petersson metric.
The study classifies weakly almost Fuchsian manifolds and proves geometric properties.
problem Classifying and understanding weakly almost Fuchsian manifolds.
method Geometric analysis and compactification techniques.
result Uniform upper bounds on volume and Hausdorff dimension for limit sets.
Holomorphic connections found on Riemann surfaces with Fuchsian monodromy.
problem Existence of holomorphic connections with Fuchsian monodromy on Riemann surfaces.
method Construction of compact Riemann surfaces with specific holomorphic vector bundles and connections.
result Existence of holomorphic connections with maximal Euler class and Fuchsian monodromy.
Study flute surfaces and Loch Ness monster, proving their parabolicity and uniformization.
problem Characterizing parabolicity and uniformization of flute surfaces and the Loch Ness monster.
method Associate sequences to Fuchsian groups and analyze their properties.
result Zero-twist flute surfaces are parabolic if and only if the series diverges.
The work is motivated by a result of Manin, which relates the Arakelov Green function on a compact Riemann surface to configurations of geodesics in a 3-dimensional hyperbolic handlebody with Schottky uniformization, having the Riemann surface as conformal boundary at infinity. A natural question is to what extent the …
In a recent paper, Q. Mérigot proved that representations in SO(2,n) of uniform lattices of SO(1,n) which are Anosov in the sense of Labourie are quasi-Fuchsian, i.e. are faithfull, discrete, and preserve an acausal subset in the boundary of anti-de Sitter space. In the present paper, we prove the reverse implication. …
Classical Dedekind sums are connected to the modular group through the construction of a (Dedekind) symbol on the cusp set of the modular group. In this paper we study generalizations of Dedekind symbols and sums that can be associated to certain Fuchsian groups uniformizing 1-punctured tori.
Paper introduces a new metric for deforming surfaces with parabolics.
problem Deformation spaces of quasifuchsian groups with parabolics.
method Developed a mapping class group invariant pressure metric on QF(S).
result Hausdorff dimension of limit sets varies analytically over QF(S).
The paper proves properties of a hyperkähler moduli space for surfaces with genus ≥ 2.
problem The hyperkähler moduli space of almost-Fuchsian structures on surfaces.
method Proofs using Teichmüller theory, Higgs bundles, and hyperbolic geometry.
result The hyperkähler structure on the moduli space extends the Weil-Petersson metric and Goldman symplectic structure.
We show that two uniform lattices of a regular right-angled Fuchsian building are commensurable, provided the chamber is a polygon with at least six edges. We show that in an arbitrary Gromov-hyperbolic regular right-angled building associated to a graph product of finite groups, a uniform lattice is commensurable with…
In this paper, it is shown that every closed hyperbolic 3-manifold contains an immersed quasi-Fuchsian closed subsurface of odd Euler characteristic. The construction adopts the good pants method, and the primary new ingredient is an enhanced version of the connection principle, which allows one to connect any two fram…
Slim curves on 3-sphere help spherical CR uniformizations.
problem Understanding curves on 3-sphere for CR uniformizations.
method Defining slimness, analyzing foliations, and applying to quasi-Fuchsian groups.
result Slim curves lead to spherical CR uniformizations of certain 3-manifolds.
Random walk on hyperbolic surfaces yields uniform distribution of hyperbolic elements.
problem Distribution of lengths in random walks on Fuchsian groups.
method Proof using Gromov's theorem on translation lengths of Gromov-hyperbolic groups.
result Geometric lengths follow laws of large numbers, central limit theorem, etc.
Method calculates systolic length of modular curves.
problem Computing upper bounds on systolic length of Riemann surfaces.
method Using congruence subgroups of hyperbolic triangle groups and traces of generators.
result Systolic length grows logarithmically with genus.
The paper proves CMC foliations for quasi-Fuchsian manifolds near the Fuchsian locus.
problem Existence of monotone CMC foliations for quasi-Fuchsian manifolds.
method Analyzes quasi-Fuchsian manifolds near the Fuchsian locus and proves the existence of a unique monotone CMC foliation.
result Proves the existence of a unique monotone CMC foliation for quasi-Fuchsian manifolds in a small neighborhood of the Fuchsian locus.
Article shows AdS-quasi-Fuchsian groups' limit sets are not smooth.
problem Smoothness of limit sets of AdS quasi-Fuchsian groups.
method Analyzes limit sets of AdS quasi-Fuchsian groups in PO(n,2).
result Limit sets are never C^1, except for Fuchsian groups.
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.
problem Determining foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus.
method Inspired by Bonahon's method, uses measured bending laminations on the boundary of convex cores.
result Measured foliations at infinity of quasi-Fuchsian manifolds can be uniquely realized for small t. Study foliations at infinity and constant mean curvature surfaces in quasi-Fuchsian manifolds.
problem Understanding foliations at infinity and constant mean curvature surfaces in quasi-Fuchsian manifolds.
method Using measured foliations and quasi-Fuchsian manifolds, proving the existence and uniqueness of foliations by constant mean curvature surfaces.
result For quasi-Fuchsian manifolds close to the Fuchsian locus, measured foliations at infinity can be uniquely realized and foliated by constant mean curvature surfaces.
We construct a Teichmueller curve uniformized by the Fuchsian triangle group (m,n,\infty) for every m<n. Our construction includes the Teichmueller curves constructed by Veech and Ward as special cases. The construction essentially relies on properties of hypergeometric differential operators. For small m, we find Bill…
Surveying Hitchin representations of Fuchsian groups.
problem Understanding representations of Fuchsian groups.
method Survey and conjectural geometric description.
result Conjectural geometric picture of an augmented Hitchin component.
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…
Measuring foliations at infinity of quasi-Fuchsian manifolds, proving filling pairs and showing realisation.
problem Understanding foliations at the boundary of quasi-Fuchsian manifolds.
method Proving filling pairs and showing realisation of measured foliations.
result Proved that measured foliations at infinity of quasi-Fuchsian manifolds are filling when close to being Fuchsian.
Study Liouville action on quasi-Fuchsian groups, proving formulas and relating to holography.
problem Analyzing Liouville action for quasi-Fuchsian groups with different types of elements.
method Derived formulas for classical Liouville action, proved first and second variations, and established holography principle.
result Established an equality linking Liouville action and renormalized volume for quasi-Fuchsian groups.
Study Fuchsian loci in mPSLn(R)-Hitchin components of a pair of pants.
problem Understanding Fuchsian loci in mPSLn(R)-Hitchin components. method Using Bonahon-Dreyer parametrization, explicit parametrization of Fuchsian loci of a pair of pants.
result Explicit parametrization of Fuchsian loci of a pair of pants.
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…
Locally rigid groups from 5-polytopes with Fuchsian ends.
problem Constructing locally rigid right-angled Coxeter groups with Fuchsian ends.
method Constructing a right-angled 5-polytope P and analyzing its right-angled Coxeter groups.
result Locally rigid right-angled Coxeter groups with Fuchsian ends can be constructed.
We prove that the renormalized volume of almost-Fuchsian hyperbolic 3-manifolds is non-negative, with equality only for Fuchsian manifolds.
Unique compact Fuchsian manifolds with convex boundary are determined by their boundary.
problem Identifying compact Fuchsian manifolds with convex boundaries.
method Proving uniqueness based on the induced path metric on the boundary.
result Compact Fuchsian manifolds with convex boundaries are uniquely determined by the induced path metric on the boundary.
The paper proves a mapping from a space of holonomy varieties to Teichmüller spaces, with a non-empty discrete intersection.
problem Intersection of Poincaré holonomy varieties and their properties.
method Holomorphic mapping and branched covering proof.
result Intersection of arbitrary Poincaré holonomy varieties is a non-empty discrete set.
We compare critical exponent for quasi-Fuchsian groups acting on the hyperbolic 3-space, H3, and on invariant disks embedded in H3. 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…
Computes cohomology groups for NEC groups, focusing on Fuchsian groups.
problem Understanding the cohomology of non-Euclidean crystallographic groups.
method Computes cohomology groups for geometrically finite NEC groups, and determines the ring structure for Fuchsian groups.
result Determination of cohomology groups and ring structures for Fuchsian groups.
The paper confirms a conjecture about foliating almost Fuchsian manifolds with CMC surfaces.
problem Confirming a conjecture about foliating almost Fuchsian manifolds with CMC surfaces.
method Proving the long-time existence and convergence of a modified mean curvature flow.
result The CMC foliation conjecture is confirmed for a subclass of almost Fuchsian manifolds.
Study on Cheeger constant in specific hyperbolic 3-manifolds.
problem Analyzing Cheeger constant in convex co-compact hyperbolic 3-manifolds.
method Examining Cheeger constant as a functional in the space of specific hyperbolic 3-manifolds.
result Global maximum of Cheeger constant is uniquely attained at the Fuchsian locus.
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) …
New findings on Fuchsian groups and their axes.
problem Determining the commensurability class of Fuchsian groups.
method Analyzing geodesic intersections and using a new approach.
result The conjecture holds for almost all Fuchsian groups.
Random walks on Fuchsian Schottky groups have harmonic measures with lower dimension.
problem Understanding the dimensionality of harmonic measures for random walks.
method Analyzing finite range random walks on Fuchsian Schottky groups.
result Harmonic measures have dimension strictly less than the limit set's Hausdorff dimension.
Study C-Fuchsian subgroups of non-arithmetic lattices.
problem Understand structure and fundamental domains of C-Fuchsian subgroups. method General procedure to analyze structure and show fundamental domains lie on a complex geodesic.
result Fundamental domains of C-Fuchsian subgroups lie on a complex geodesic homeomorphic to the unit disk. Alternative construction of quasi-Fuchsian flows using vortex equations.
problem Constructing quasi-Fuchsian flows.
method Using coupled vortex equations to construct quasi-Fuchsian flows as thermostats.
result Formulas for the marked length spectrum of quasi-Fuchsian flows.
Totally geodesic limit of quasi-Fuchsian groups with converging critical exponent.
problem Understanding convergence of quasi-Fuchsian groups.
method Critical exponent convergence analysis.
result Convergence of quasi-Fuchsian groups to totally geodesic representations.
The paper shows nearly-Fuchsian properties for certain hyperbolic 3-manifolds.
problem Characterizing hyperbolic 3-manifolds with specific surface properties.
method Analyzing minimal and non-minimal surfaces with principal curvatures in [-1,1](-1,1).
result Weakly almost-Fuchsian manifolds are nearly-Fuchsian.
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 trace set of a Fuchsian group Γ ist the set of length of closed geodesics in the surface Γ\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.
problem Understanding proper affine actions of surface groups.
method Explicit neighborhoods in quasifuchsian space and critical points of entropy.
result Critical points of entropy lie on the Fuchsian locus.