The paper studies hyperbolic quotients of projection complexes and their actions.
problem Understanding the structure and properties of quotients of projection complexes.
method Analyzing the quotient of projection complexes by normal subgroups and studying the resulting actions.
result The quotient complex is δ-hyperbolic under certain conditions, and the quotient group is acylindrically hyperbolic.
Study semiclassical measures on complex hyperbolic quotients, identifying measure supports.
problem Understanding Laplacian eigenfunctions on complex hyperbolic quotients.
method Combining fractal uncertainty principle and Ratner theory to analyze measure supports.
result Semiclassical measures support is either cosphere bundle or a compact submanifold.
The study shows how quotients of mapping class groups are hierarchically hyperbolic.
problem Understanding the hierarchical hyperbolicity of mapping class groups and their quotients.
method A combinatorial criterion for hierarchical hyperbolicity applied to mapping class groups.
result Quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic.
Random quotients preserve hyperbolic properties in groups.
problem Preserving hyperbolic properties in random group quotients.
method Independent random walks, spinning families, projection complexes.
result Random quotients of acylindrical and hierarchical hyperbolic groups remain so.
New insights into a complex hyperbolic braid group quotient.
problem Understanding a complex hyperbolic braid group quotient.
method Analyzing the moduli space of 12-tuples in CP1 and identifying loops.
result Identifying loops in the 9-ball quotient corresponding to standard braid generators.
Study on curves on specific arithmetic quotients of hyperbolic 2-ball.
problem No complex curves of certain genus on these arithmetic quotients.
method Volume estimates and understanding special subvarieties.
result For large discriminants, no complex curves of fixed genus.
New method constructs non-arithmetic hyperbolic orbifolds from complex arithmetic ball quotients.
problem Creating non-arithmetic lattices in projective orthogonal groups.
method Using anti-holomorphic involutions on complex arithmetic ball quotients, gluing fixed loci along geodesic subspaces.
result Explicit calculation of the volume of constructed non-arithmetic orbifolds.
Study on curvature properties and Shafarevich conjecture for complex hyperbolic manifolds.
problem Existence and nonexistence of Kähler metrics with nonpositive curvature on toroidal compactifications.
method Analysis of toroidal compactifications of finite volume complex hyperbolic manifolds, verification of Shafarevich conjecture.
result Verification of Shafarevich conjecture for compactifications of quotients of complex hyperbolic space by non-uniform arithmetic lattices.
New theorem bounds group quotient size to subgroups index.
problem Understanding subgroup structure in hyperbolic groups.
method Proved quotient size bounds on Eilenberg-MacLane spaces.
result One-ended hyperbolic groups cannot have isomorphic finite-index subgroups.
Study Bergman kernels on complex hyperbolic cusps, generalizing previous results.
problem Localization of Bergman kernels on Kähler manifolds with complex hyperbolic cusps.
method Revisiting Tian's peak section method, applying to Kähler-Einstein metrics and quotients of complex balls.
result Partial localization result for Poincaré type cusps.
We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic 13-space, take the quotient of the remaining space by a discrete group, and find generators for the orbifold fundamental group of the quotient. These generators have the most natural fo…
Study cohomology of ball quotients and their compactifications.
problem Cohomology of symmetric power of cotangent bundles of ball quotients and their compactifications.
method Hodge theory for complete hermitian manifolds, Green's operator, extension of results.
result Established existence of Hodge decomposition and Green's operator for ball quotients and their compactifications.
The paper establishes analogs of Stallings' theorem for group homomorphisms and their nilpotent quotients.
problem Understanding the structure of fundamental groups of geometric objects.
method Develops analogs of Stallings' theorem for group homomorphisms and their nilpotent quotients.
result Derives applications including non-isomorphic number fields and hyperbolic manifolds with isomorphic universal nilpotent quotients.
New 2D complex hyperbolic structures found on sphere orbibundles.
problem Locally rigid complex hyperbolic structures on sphere orbibundles.
method Constructing families of complex hyperbolic structures on disc orbibundles.
result Examples of non-locally rigid complex hyperbolic structures.
We present a new criterion for the complex hyperbolicity of a non-compact quotient X of a bounded symmetric domain. For each p ≥ 1, this criterion gives a precise condition under which the subvarieties V ⊂ X with dim V ≥ p are of general type, and X is p-measure hyperbolic. Then, we give several applica…
Random quotients of hyperbolic cubulated groups remain cubulated.
problem Understanding properties of random quotients of hyperbolic cubulated groups.
method Cubical small-cancellation theory, exponential growth of conjugacy classes, and hyperplane stabilizers' growth.
result Low-density random quotients of cubulated hyperbolic groups are cubulated and hyperbolic.
Study of Dehn filling quotients in hierarchically hyperbolic groups.
problem Understanding the structure of Dehn filling quotients in specific groups.
method Introduced a construction for cusped spaces of relatively hyperbolic groups and used it to study Dehn-filling-like quotients.
result Infinite hyperbolic quotients of mapping class groups of punctured spheres and braid groups are found.
New group constructed from cube complex properties.
problem Creating a new group from cube complex properties.
method Cubical Rips construction for finitely presented groups.
result New group surjects onto a given group with specific properties.
In 1985 D.Sullivan had introduced a dictionary between two domains of complex dynamics: iterations of rational functions on the Riemann sphere and Kleinian groups. The latters are discrete subgroups of the group of conformal automorphisms of the Riemann sphere. This dictionary motivated many remarkable results in both …
The paper studies binary icosahedral representations of hyperbolic 3-manifolds.
problem Understanding the representations of hyperbolic integral homology spheres into the binary icosahedral group.
method Relating 2I representations to quotient dimension and analyzing finite covers. result Hyperbolic 3-manifolds have quotient dimension 2 or 3, with specific cases obtained infinitely many times.
New Kähler manifolds found with nonpositive curvature operators.
problem Rigidity of curvature operators in Kähler manifolds.
method Proved existence of Kähler manifolds with specific curvature properties.
result Kähler manifolds can have nonpositive curvature operators, unlike quaternionic and Cayley hyperbolic manifolds.
Knot groups of hyperbolic 2-bridge knots are uniquely identified by their finite quotients.
problem Identifying knots based on their group structures.
method Proving hyperbolic 2-bridge knots are uniquely determined by their profinite completions.
result Hyperbolic 2-bridge knots are uniquely identified by their finite quotients.
The paper studies acylindrical actions on trees and proves acylindrical hyperbolicity of Baumslag-Solitar groups.
problem Exploring acylindrical actions on trees and their properties.
method Demonstrates criteria for preserving acylindrical hyperbolicity and analyzes the outer automorphism group of Baumsligar-Solitar groups.
result Proves acylindrical hyperbolicity of non-solvable Baumsligar-Solitar groups.
We study forgetful maps between Deligne-Mostow moduli spaces of weighted points on P^1, and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livné fibrations and the Mostow/Toledo maps between complex hyperbolic surfaces. They also in…
We study the classification of smooth toroidal compactifications of nonuniform ball quotients in the sense of Kodaira and Enriques. Moreover, several results concerning the Riemannian and complex algebraic geometry of these spaces are given. In particular we show that there are compact complex surfaces which admit Riem…
Finite quotients of fibered hyperbolic 3-manifold groups detect taut polynomials.
problem Detecting taut polynomials of fibered faces of Thurston norm balls
method Developing a framework for profinite invariance of twisted multivariable Alexander polynomials
result Proving finite quotients detect taut polynomials
In this paper, we study punctured spheres in two dimensional ball quotient compactifications (X,D). For example, we show that smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded 3-punctured spheres. We also use totally geodesic punctured spheres to prove ampleness o…
Compact quotients of homogeneous spaces are studied, leading to new findings about sphere bundles.
problem Understanding compact quotients of reductive homogeneous spaces and their implications.
method Analyzing normal bundles and sphere bundles associated with these spaces, proving homotopy triviality conditions.
result Many reductive homogeneous spaces do not admit compact quotients, resolving conjectures.
Researchers found the global topology of the Eisenstein-Picard modular surface.
problem Understanding the global topology of the Eisenstein-Picard modular surface.
method Quotient space of the complex hyperbolic plane by the modular group.
result Determined the global topology of the Eisenstein-Picard modular surface as a 4-orbifold.
Study of subgroups in complex hyperbolic lattice triangle groups.
problem Characterizing subgroups of finite index in complex hyperbolic lattice triangle groups.
method Explicit construction and analysis of subgroups, examination of their properties.
result Identification of neat subgroups, subgroups with positive first Betti number, and homomorphisms onto non-Abelian free groups.
We investigate relation between Dehn fillings and commensurability of hyperbolic 3-manifolds. The set consisting of the commensurability classes of hyperbolic 3-manifolds admits the quotient topology induced by the geometric topology. We show that this quotient space satisfies some separation axioms. Roughly speaking, …
In a Riemannian manifold a regular convex domain is said to be λ-convex if its normal curvature at each point is greater than or equal to λ. In a Hadamard manifold, the asymptotic behaviour of the quotient $\vol(Ω(t))/\vol(\partialΩ(t))$ for a family of λ-convex domains Ω(t) expanding over the whole space has b…
In this paper, we study the geometry of cone-offs of CAT(0) cube complexes over a family of combinatorially convex subcomplexes, with an emphasis on their Gromov-hyperbolicity. A first application gives a direct cubical proof of the characterization of the (strong) relative hyperbolicity of right-angled Coxeter groups,…
We define the Kobayashi quotient of a complex variety by identifying points with vanishing Kobayashi pseudodistance between them and show that if a compact complex manifold has an automorphism whose order is infinite, then the fibers of this quotient map are nontrivial. We prove that the Kobayashi quotients associated …
Using Legendrian immersions and, in particular, Legendre curves in odd dimensional spheres and anti De Sitter spaces, we provide a method of construction of new examples of Hamiltonian-minimal Lagrangian submanifolds in complex projective and hyperbolic spaces, including explicit one parameter families of embeddings of…
This paper completes a classification of the types of orientable and non-orientable cusps that can arise in the quotients of hyperbolic knot complements. In particular, S2(2,4,4) cannot be the cusp cross-section of any orbifold quotient of a hyperbolic knot complement. Furthermore, if a knot complement covers an orb…
The study shows a finite number of groups acting on hyperbolic spaces with bounded entropy and compact quotient.
problem Finite number of groups acting on hyperbolic spaces with bounded entropy and compact quotient.
method Analyzing torsion-free groups acting by isometries on hyperbolic metric spaces with bounded entropy and compact quotient.
result The set of such groups is finite and can be estimated based on hyperbolicity constant, entropy, and quotient diameter.
We discuss a problem posed by Gersten: Is every automatic group which does not contain Z+Z subgroup, hyperbolic? To study this question, we define the notion of "n-tracks of length n", which is a structure like Z+Z, and prove its existence in the non-hyperbolic automatic groups with mild conditions. As an application, …
On manifolds with an even Riemannian conformally compact Einstein metric, the resolvent of the Lichnerowicz Laplacian, acting on trace-free, divergence-free, symmetric 2-tensors is shown to have a meromorphic continuation to the complex plane, defining quantum resonances of this Laplacian. For higher rank symmetric ten…
The paper studies surface quotients of Fuchsian buildings.
problem Understanding group actions and symmetries in Fuchsian buildings.
method Developed theory of surface quotients, proved existence of discrete subgroups.
result Existence of discrete subgroups whose quotient is a compact surface.
Let G be a simple complex Lie group, $\alg{g}$ be its Lie algebra, K be a maximal compact form of G and $\alg{k}$ be a Lie algebra of K. We denote by X→X the anti-involution of $\alg{g}$ which singles out the compact form $\alg{k}$. Consider the space of flat $\alg{g}$-valued connections…
New examples of hyperbolic 3-manifolds with unique profinite structure.
problem Finding hyperbolic 3-manifolds with unique profinite structure.
method Examining fundamental groups of closed fibered hyperbolic 3-manifolds.
result First examples of closed fibered hyperbolic 3-manifolds with unique profinite structure.
The bending map of a hyperbolic 3-manifold maps a convex cocompact hyperbolic metric on a hyperbolic 3-manifold with boundary to its bending measured geodesic lamination. In the present paper we study the extension of this map to the space of geometrically finite hyperbolic metrics. We introduce a relationship on the s…
In this paper we prove that for all n=4k−2, k≥2 there exists a closed smooth complex hyperbolic manifold M with real dimension n having non-trivial π1(T<0(M)). T<0(M) denotes the Teichmüller space of all negatively curved Riemannian metrics on M, which is the topological quoti…
Random quotients of mapping class groups have rigid properties.
problem Rigidity of random quotients of mapping class groups.
method Generalization of Ivanov's theorem and use of hierarchically hyperbolic groups.
result Automorphisms and commensurators of random quotients coincide with the groups themselves.
Quotients of Gordian and H(2)-Gordian graphs are hyperbolic.
problem Investigate quotients of Gordian and H(2)-Gordian graphs under knot invariants.
method Defined equivalence relations by knot invariants (det, Jones span, tricolorability) and showed quotient graphs are Gromov hyperbolic.
result Quotients of H(2)-Gordian graph of links modulo span of Jones polynomial is isomorphic to complete graph.
The study distinguishes knots using finite quotients of their fundamental groups.
problem Distinguishing knots in 3-dimensional space.
method Using finite quotients of the fundamental group of knot complements.
result The method successfully distinguishes specific knots.
The study examines power quotients of surface groups and mapping class groups, proving structural properties and isomorphisms.
problem Structural properties and isomorphisms of power quotients of surface groups and mapping class groups.
method Analyzes the outer automorphism and automorphism groups of power quotients, proving isomorphisms and structural properties.
result The outer automorphism group of Γ(n) is isomorphic to the quotient of the extended mapping class group of S by nth powers of Dehn twists.