Researchers prove the arc complexes of decorated hyperbolic polygons are balls.
problem Understanding the structure of decorated hyperbolic polygons.
method Combinatorial approach using pseudo-manifolds and shellability.
result Arc complexes of decorated hyperbolic polygons are closed piecewise linear balls.
Study strip deformations of hyperbolic polygons with decorated vertices.
problem Understanding deformations of hyperbolic polygons with decorated vertices.
method Analyzing strip deformations of ideal hyperbolic polygons with horoballs.
result Arc complexes parameterize uniformly lengthening deformations.
This paper studies deformations of hyperbolic surfaces with special structures.
problem Infinitesimal deformations of hyperbolic surfaces with boundary and ideal vertices.
method Description of the admissible cone of deformations in terms of the arc complex.
result Realization of the admissible cone and its faces as arc complexes for specific surface families.
Researchers extend parametrization of Margulis spacetimes using strip deformations.
problem Parametrize Margulis spacetimes with decorated horoballs.
method Use strip deformations to parametrize complete finite-area hyperbolic surfaces with spikes decorated with horoballs.
result Generalized parametrization of Margulis spacetimes with photons.
Study Liouville theorem for specific harmonic maps.
problem Proving Liouville theorem for harmonic maps.
method Establish gradient estimates under specific conditions.
result Proved Liouville theorem for harmonic maps.
Let M be a complete simply connected Riemannian manifold, with sectional curvature K bounded above by -1. Under some assumptions on the geometry of the boundary of M, which are satisfied for instance if M is a symmetric space, or has dimension 2, we prove that given any family of horoballs in M, and any point x_0 outsi…
Study of flow in hyperbolic space with capillary boundary.
problem Optimizing hypersurfaces with capillary boundary in hyperbolic space.
method Mean curvature flow with capillary boundary, preserving volume and energy.
result Flow converges to a truncated umbilical hypersurface.
Study on hyperbolic manifolds with special boundaries.
problem Characterizing geometric properties of hyperbolic manifolds with specific boundaries.
method Analyzing asymptotically hyperbolic manifolds with a horospherical boundary.
result Derived geometric formulas for the model case of horoballs and their complements.
New proof for hyperbolic groups using contracting boundaries of cusped spaces.
problem Proving groups are hyperbolic relative to subgroups.
method Proving compact contracting boundary of cusped space implies hyperbolicity.
result If cusped space Gh has compact contracting boundary, then G is hyperbolic relative to subgroups H. Proves symmetric spaces for certain quasi-isometric properties.
problem Understanding quasi-isometric spaces and their properties.
method Analyzes geometric and geometrically finite quasi-actions.
result Proves quasi-isometric spaces are symmetric or real line.
Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. We study the asymptotics of the number of geodesics in M starting from and returning to a given cusp, and of the number of horoballs at parabolic fixed points in the universal cover of M. In the appendix, due to K. Bel…
We prove mean curvature estimates and a Jorge-Koutroufiotis type theorem for submanifolds confined into either a horocylinder of N X L or a horoball of N, where N is a Cartan-Hadamard manifold with pinched curvature. Thus, these submanifolds behave in many respects like submanifolds immersed into compact balls and into…
In the course of our work on low-volume hyperbolic 3-manifolds, we came upon a linking problem for horoball necklaces in H3. A horoball necklace is a collection of sequentially tangent beards (i.e. spheres) with disjoint interiors lying on a flat table (i.e. a plane) such that each bead is of diameter at mo…
Given a discrete subgroup of the isometries of n-dimensional hyperbolic space there is always a region kept precisely invariant under the stabilizer of a parabolic fixed point, called the Margulis region. While in dimensions 2 and 3 this region is a horoball, it has in general a more complicated shape due to the existe…
Study of quantum decorated character stacks and their quantizations.
problem Quantization of decorated character stacks and their compatibility with cutting and gluing.
method Using stratified factorization homology, extend Fock and Goncharov's construction to include stacky points.
result Construction of categorical charts and flips on quantum decorated character stacks.
We study surfaces with decorations and prove uniformization in non-Euclidean geometries.
problem Discrete conformal equivalence in non-Euclidean geometries.
method Variational principle and continuous deformation.
result One master theory of discrete conformal equivalence across different geometries.
Minimal non-arithmetic hyperbolic 3-orbifold found with least volume.
problem Finding the hyperbolic 3-orbifold with minimal volume among non-arithmetic ones.
method Utilized the tetrahedral Coxeter group and horoball configuration to prove minimal volume.
result The 1-cusped quotient of hyperbolic space by the tetrahedral Coxeter group has minimal volume.
Decorated TQFTs compute invariants with additional structures.
problem Computing topological invariants with additional structures.
method Cutting and gluing to obtain decorated invariants, proposing Hilbert spaces for two-dimensional surfaces.
result Proposal for Hilbert spaces assigned to surfaces in decorated TQFTs.
Researchers created group presentations for specific Bianchi groups.
problem Finding explicit group presentations for certain Bianchi groups.
method Applied Macbeath's theorem to a large horoball of the Bianchi groups as discrete subgroups of H3. result Produced a complete list of group presentations for specified Bianchi groups.
This study extends a result on quasi-isometry of hyperbolic groups to relatively hyperbolic groups.
problem Classifying groups up to quasi-isometry, focusing on relatively hyperbolic groups.
method Defining quasiconformal maps and showing their equivalence to coarsely cusp-preserving quasi-isometries between Bowditch boundaries.
result Quasiconformal maps between Bowditch boundaries of relatively hyperbolic groups are equivalent to coarsely cusp-preserving quasi-isometries.
Constructs TQFTs for cobordisms with cohomology class decorations.
problem Creating TQFTs for cobordisms with cohomology class decorations.
method Starting from an abelian group G and a factorizable ribbon Hopf G-bialgebra H, constructs a TQFT JH for connected framed cobordisms between connected surfaces with connected boundary decorated with cohomology classes with coefficients in G. result Our functor recovers a special case of Kerler-Lyubashenko TQFTs when restricted to trivial decorations.
Study compares constrained and decoupled moduli spaces of manifolds with particles and discs.
problem Comparing constrained and decoupled moduli spaces of manifolds with embedded particles and discs.
method Generalized Bödigheimer--Tillmann's work to higher dimensions and different tangential structures.
result New results for surfaces with different tangential structures and higher dimensional manifolds.
Study local features of decorated representation spaces for spherical surfaces.
problem Local structure of moduli space of spherical surfaces with conical points.
method Analysis of decorated representation spaces of fundamental groups in SU(2).
result Smooth locus of decorated representation spaces is dense and connected.
The article describes how decorations on hyperbolic surfaces lead to unique tessellations and decompositions.
problem Understanding the geometric structure of decorated hyperbolic surfaces.
method Developing a characterisation of canonical tessellations and dual decompositions using hyperbolic geometry.
result Decorations on hyperbolic surfaces induce unique canonical tessellations and dual decompositions.
Corrects a 1-off error in Harer's spine dimension calculation for decorated Teichmüller spaces.
problem Incorrect dimension calculation in Harer's spine for decorated Teichmüller spaces.
method Identifies and corrects the dimension discrepancy in Harer's spine construction.
result Corrects the dimension of Harer's spine by 1 for decorated Teichmüller spaces.
This is a survey on the project `Decorated Marked Surfaces', where we introduce the decoration Δ on a marked surfaces S, to study Calabi-Yau-2 (cluster) categories, Calabi-Yau-3 (Fukaya) categories, braid groups for quivers with potential, quadratic differentials and stability conditions.
DecoR estimates causal effects in confounded time series data.
problem Estimating causal effects in time series with unobserved confounders.
method Robust regression in the frequency domain.
result Proves upper bounds for estimation error of DecoR, implying consistency.
Enhanced Teichmüller space for surfaces with decorations and enhancements.
problem Parameterizing and understanding Teichmüller spaces with enhancements and decorations.
method Introduced a new variation of Teichmüller space, constructed parameterization, and introduced lamination space.
result Compatibility of shear coordinates and λ-length coordinates in the new deformation space.
Unified framework for smooth structures on coadjoint orbits.
problem Smooth structures on coadjoint orbits of diffeomorphism groups.
method Decorated and augmented nonlinear Grassmannians, functors, smooth structure.
result Uniform description of coadjoint orbits' smooth structures.
Study of decorated surfaces with vortices and their group structures.
problem Understanding group structures of decorated surfaces with vortices.
method Proved isomorphism between cluster braid group, braid twist group, and fundamental group of moduli space.
result Finite presentations of isomorphic groups were given.
Discrete conformal maps on surfaces with vertex decorations are studied.
problem Discrete conformal equivalence for decorated piecewise Euclidean surfaces.
method Intimate relationship between decorated PE-surfaces, canonical tessellations of hyperbolic surfaces, and convex hyperbolic polyhedra; concave variational principle.
result Proof of discrete uniformization theorem for decorated PE-surfaces.
New TQFT for link cobordisms without decoration.
problem Tackles the problem of defining a TQFT for link cobordisms.
method Introduces a Heegaard-Floer homology functor.
result Independently constructs a TQFT for link cobordisms.
We produce a one-parameter family of coordinates {Ψh}h∈R of the decorated Teichmüller space of an ideally triangulated punctured surface (S,T) with negative Euler characteristic, which is a deformation of Penner's simplicial coordinate \cite{P1}. If h⩾0, the decorated Teichmüller space in…
We are interested in the 3-Calabi-Yau categories D arising from quivers with potential associated to a triangulated marked surface S (without punctures). We prove that the spherical twist group ST of D is isomorphic to a subgroup (generated by braid twists) of the mapping class group …
Let S be a path-connected, locally-compact CW-complex, and let M be a subcomplex with finitely-many components. A `decorated SL_2(C)-local system' is an SL_2(C)-local system on S, together with a choice of `decoration' at each component of M (a section of the stalk of an associated vector bundle). We study the (decorat…
The first aperiodic monotiling, introduced by Taylor, was based on a trapezoidal prototile equipped with 14 distinct decorations. A presentation of the closely related Taylor-Socolar aperiodic monotiling is based on a hexagonal prototile equipped with 7 decorations. This paper gives decoration-free algebraic descriptio…
We give a finite presentation for the braid twist group of a decorated surface. If the decorated surface arises from a triangulated marked surface without punctures, we obtain a finite presentation for the spherical twist group of the associated 3-Calabi-Yau triangulated category. The motivation/application is that the…
The main goal is to find the Homfly polynomial of a link formed by decorating each component of the Hopf link with the closure of a directly oriented tangle. Such decorations are spanned in the Homfly skein of the annulus by elements Q_λ, depending on partitions λ. We show how to find the 2-variable Homfly invariant <λ…
An unknotting tunnel in a 3-manifold with boundary is a properly embedded arc, the complement of an open neighborhood of which is a handlebody. A geodesic with endpoints on the cusp boundary of a hyperbolic 3-manifold and perpendicular to the cusp boundary is called a vertical geodesic. Given a vertical geodesic in a h…
The paper proves a theorem for discretizing Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with nonpositive Euler number.
method Discrete conformal theory and variational principles with constraints.
result Each decorated piecewise Euclidean metric on surfaces with nonpositive Euler number is discrete conformal to a metric with a specific discrete curvature constant.
Generalising a seminal result of Epstein and Penner for cusped hyperbolic manifolds, Cooper and Long showed that each decorated strictly convex projective cusped manifold has a canonical cell decomposition. Penner used the former result to describe a natural cell decomposition of decorated Teichmüller space of puncture…
The punctured solenoid § is an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichmüller space of § is introduced, studied, and found to be parametrized by certain coordinates on a fixed triangulation of §. Furthermore…
Extends Paulin's result to relatively hyperbolic groups.
problem Proving quasi-isometric equivalence between relatively hyperbolic groups.
method Introducing relative quasi-Mobius maps and using coarsely cusp-preserving quasi-isometries.
result Establishes a homeomorphism between Bowditch boundaries inducing quasi-Mobius maps.
Study on hidden symmetries in Dehn fillings of tetrahedral links.
problem Existence of infinite families of knot complements with hidden symmetries.
method Analysis of horoball packings and SnapPy algorithm.
result No infinite families of knot complements with hidden symmetries exist for given tetrahedral links and related manifolds.
This paper extends the decorated Teichmüller theory developed before for punctured surfaces to the setting of ``bordered'' surfaces, i.e., surfaces with boundary, and there is non-trivial new structure discovered. The main new result identifies the arc complex of a bordered surface up to proper homotopy equivalence wit…
The paper introduces a new discretization of Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with conic singularities.
method Discrete conformal theory and variational principles with constraints.
result Established a discrete uniformization theorem for surfaces with non-positive Euler number.
We study a new bordification of the decorated Teichmüller space for a multiply punctured surface F by a space of filtered screens on the surface that arises from a natural elaboration of earlier work of McShane-Penner. We identify necessary and sufficient conditions for paths in this space of filtered screens to yield …
Study rigidity and volume optimization of hyperbolic polyhedra.
problem Rigidity and volume optimization of hyperbolic polyhedra.
method Analyzing decorated 1-3 type hyperbolic polyhedra and their metrics.
result Decorated 1-3 type hyperbolic polyhedra are rigid up to isometry and change of decorations.