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.
Study quotients of curve complex actions by mapping class group.
problem Understanding actions of mapping class group on curve complex quotients.
method Cone off uniformly quasi-convex subspaces to form symmetric curve sets, non-maximal train track sets, and compression body disc sets. Analyze actions of mapping class group on these quotients.
result Actions of mapping class group on quotients are strongly WPD, non-elementary, and have infinite diameter.
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.
Study centers of mapping-torus groups to define knot and mapping class invariants.
problem Understanding the center of mapping-torus groups.
method Determine the center of meta-nilpotent quotients of mapping-torus groups.
result Introduce two invariants of knots and mapping classes as quadratic forms.
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.
The paper explores mapping class group quotients by Dehn twists and their representations.
problem Finite quotients and representations of mapping class groups by powers of Dehn twists.
method Construction of finite quotients using representations with Zariski dense images into semisimple Lie groups, and Long and Moody's method.
result The Fibonacci TQFT representation is a specialization of the Jones representation in genus 2.
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. Investigates intrinsic Lipschitz sections in nonlinear quotient maps.
problem Analyzing intrinsic Lipschitz sections in non-linear quotient maps.
method Introduced Leibniz formula for intrinsic slope under weaker conditions, used properties of intrinsic dilations in Carnot groups, and provided conditions for sum of sections.
result Found conditions for sum of intrinsically Lipschitz sections in Carnot groups of step 2.
Study compares Kähler quotients of torus actions under varying moment maps.
problem Comparing Kähler quotients of torus actions under varying moment maps.
method Analyzes the transformation of Kähler quotients as moment maps change, proving bimeromorphic transformations and desingularizations.
result Each nondegenerate singular Kähler quotient has a partial and rational desingularization.
In the first part of this paper we prove that the mapping class subgroups generated by the D-th powers of Dehn twists (with D≥2) along a sparse collection of simple closed curves on an orientable surface are right angled Artin groups. The second part is devoted to power quotients, i.e. quotients by the normal s…
New proofs show smallest non-cyclic quotients for braid and mapping class groups.
problem Classifying smallest non-cyclic quotients of braid and mapping class groups.
method Elementary proofs without Bertrand-Chebyshev theorem.
result Smallest non-cyclic quotients for braid and mapping class groups identified.
Defines super stable maps and proves quotient superorbifolds for genus zero.
problem Defines stable supercurves and super stable maps of genus zero.
method Uses labeled trees and slice theorem for super Lie groups.
result Proves moduli space of stable supercurves and super stable maps are quotient superorbifolds.
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…
A new presentation of a quotient of braid groups leads to a new type of Burnside group.
problem Understanding the structure of quotient groups of braid groups.
method Purely group-theoretic methods, including presentations and finiteness results.
result A new presentation for the kernel of a truncated quotient map of braid groups.
The study proves unique path lifting properties and their implications on quotient spaces and covering maps.
problem Understanding unique path lifting properties and their implications on quotient spaces and covering maps.
method The study uses group actions on R-trees and path lifting properties to prove the main results. result Every map of manifolds with the unique path lifting property is a covering map.
Given a metric space X and a function f:X→R, the Reeb construction gives metric a space Xf together with a quotient map X→Xf. Under suitable conditions Xf becomes a metric graph and can therefore be used as a graph approximation to X. The Gromov-Hausdorff distance from Xf to X is b…
The paper defines conditions for a Riemannian structure on a symplectic quotient.
problem Existence of Riemannian structures on symplectic quotients.
method Analyzes conditions for existence given a Lie group action with equivariant momentum mapping.
result Determines conditions under which an induced Riemannian structure exists.
We give a short proof of Masbaum and Reid's result that mapping class groups involve any finite group, appealing to free quotients of surface groups and a result of Gilman, following Dunfield-Thurston.
Study of one-dimensional non-Hausdorff manifolds and their quotient to CW complexes.
problem Understanding and characterizing one-dimensional non-Hausdorff manifolds.
method Analyzing properties of connected non-Hausdorff manifolds and their quotient spaces to CW complexes.
result Existence of a quotient map from a connected non-Hausdorff manifold to an open one-dimensional CW complex.
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…
Quotients of torus endomorphisms have parabolic orbifolds.
problem Understanding the structure of quotients of torus endomorphisms.
method Analyzing the properties of torus endomorphisms and their quotients.
result Every quotient of a torus endomorphism has a parabolic orbifold.
The purpose of the article is to study a foliation associated to a lattice-equivariant harmonic map of small rank from a complex ball to another. The result is related to rigidity of some complex ball quotients.
Embeds CR manifolds into complex spaces using equivariant actions.
problem Embedding strongly pseudoconvex CR manifolds into complex spaces.
method Equivariant CR maps and quotient maps.
result Universal quotient map property for CR manifolds.
The study examines the rigidity of mapping class groups under large powers of twists.
problem Quasi-isometric rigidity of mapping class groups under large powers of twists.
method Analyzing quotients of mapping class groups by large powers of Dehn twists, using techniques from hierarchically hyperbolic spaces.
result Quasi-isometric rigidity and small automorphism groups of the studied quotients.
This paper investigates which smooth manifolds arise as quotients (orbit spaces) of flows of vector fields. Such quotient maps were already known to be surjective on fundamental groups, but this paper shows that every epimorphism of countably presented groups is induced by the quotient map of some flow, and that higher…
In this paper, we investigate the existence of a subclass of quotients of affine connection control systems, which preserve the mechanical structures. Both local and global sufficient and necessary conditions are given for the geodesically accessible affine connection control systems such that they can admit this subcl…
Using quantum representations of mapping class groups we prove that profinite completions of Burnside-type surface group quotients are not virtually prosolvable, in general. Further, we construct infinitely many finite simple characteristic quotients of surface groups.
The study explores maps of 2- and 3-uniform tilings on the torus.
problem Understanding the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
method Analyzing the quotient maps of 2- and 3-uniform tilings on the torus.
result Bounds on the number of vertex orbits in quotient maps of 2- and 3-uniform tilings.
We solved a conjecture about braid group quotients being alternating groups.
problem Understanding the smallest non-trivial quotients of braid group commutator subgroups.
method Proved the conjecture about alternating groups as quotients, showed minimal quotient maps.
result Proved conjecture about braid group quotients being alternating groups.
Polynomial Duistermaat-Heckman measure on symplectic groupoid quotients.
problem Calculating measures on symplectic groupoid quotients.
method Using Hamiltonian groupoid actions and proper moment maps.
result Duistermaat-Heckman measure is polynomial.
Constructs a moment map for maps to balanced manifolds.
problem Understanding maps from complex manifolds to balanced manifolds.
method Constructs a moment map for a specific action of biholomorphisms.
result Lays groundwork for balanced quotients.
To every Q-irreducible representation r of a finite group H, there corresponds a simple factor A of Q[H] with an involution τ. To this pair (A,τ), we associate an arithmetic group Ω consisting of all (2g−2)×(2g−2) matrices over a natural order of Aop which preserve a natural skew-Hermitian …
The paper extends arithmetic quotient results to right-angled Artin groups.
problem Arithmetic quotients of automorphism groups of free groups and mapping class groups.
method Analogous methods to free groups and mapping class groups applied to right-angled Artin groups.
result New virtual arithmetic quotients of Aut(F_n) for n ≥ 4, containing nonabelian free groups.
We prove that the minimal nontrivial finite quotient group of the mapping class group M_g of a closed orientable surface of genus g is the symplectic group PSp(2g,Z_2), for g = 3 and 4 (this might remain true, however, for arbitrary genus g > 2). We discuss also some results for arbitrary genus g.
The paper examines Euclidean domains with nearly maximal Yamabe quotients.
problem Understanding domains with nearly maximal Yamabe quotients in Euclidean space.
method Analyzes the properties of domains in R3 with nearly maximal Yamabe quotients, proving conditions for equality and comparing to quasi-conformal maps. result Domains with nearly maximal Yamabe quotients are diffeomorphic to balls and are close to a ball in a metric space sense.
M-theory preons -- solutions of eleven-dimensional supergravity preserving 31 supersymmetries -- have recently been shown to be locally maximally supersymmetric. This implies that if preons exist they are quotients of maximally supersymmetric solutions. In this paper we show that no such quotients exist. This is achiev…
This note investigates the so-called Tube map which connects welded knots, that is a quotient of the virtual knot theory, to ribbon torus-knots, that is a restricted notion of fillable knotted tori in the 4-sphere. It emphasizes the fact that ribbon torus-knots with a given filling are in one-to-one correspondence with…
We determine the Lyapunov spectrum of ball quotients arising from cyclic coverings. The computations are performed by rewriting the sum of Lyapunov exponents as ratios of intersection numbers and by the analysis of the period map near boundary divisors. As a corollary, we complete the classification of commensurability…
The study examines the topology of map germs and their images.
problem Understanding the topology of map germs and their images.
method Using the topology of the link to analyze the normal and non-normal images.
result Normal images of map germs are quotient singularities.
Functoriality proved for higher rho invariants of elliptic operators.
problem Computing higher rho invariants of elliptic operators.
method Functoriality proved through finite-propagation argument.
result Maximal higher rho invariants behave functorially under quotient maps.
We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space R^{4d} by the action of a compact Abelian group. These 4n-dimensional quotients carry a multi-Hamilitonian action of an n-torus. The image of the hypersymplectic moment map for this torus action may be described by a configuration o…
This paper shows semi-equivelar toroidal maps are vertex-transitive covers.
problem Understanding the relationship between semi-equivelar and vertex-transitive toroidal maps.
method Proving semi-equivelar toroidal maps are quotients of vertex-transitive toroidal maps.
result Each semi-equivelar toroidal map has a finite vertex-transitive cover.
Let G be a complex reductive group and K a maximal compact subgroup. If X is a smooth projective G-variety, with a fixed (not necessarily integral) K-invariant Kaehler form, then the K-action is Hamiltonian. Let M be the zero fiber of the corresponding moment map. It is well known that the quotient M/K is a complex spa…
If a Lie group acts on a manifold freely and properly, pulling back by the quotient map gives an isomorphism between the differential forms on the quotient manifold and the basic differential forms upstairs. We show that this result remains true for actions that are not necessarily free nor proper, as long as the ident…
Introduces intrinsically Lipschitz graphs in metric spaces.
problem Graphs in metric spaces with Lipschitz conditions.
method Focuses on quotient maps and intrinsically Lipschitz sections.
result Compactness, Ahlfors regularity, and extension theorems.
Study the relationship between orbit braid group and equivariant mapping class group on surfaces.
problem Understanding the relationship between mapping class groups and braid groups with group actions.
method Using the fibration F0GMightarrowF(M/G,n) and exact sequence. result The conclusion is closely connected with the braid group of the quotient space.
Thurston's fibered face theory allows us to partition the set of pseudo-Anosov mapping classes on different compact oriented surfaces into subclasses with related dynamical behavior. This is done via a correspondence between the rational points on fibered faces in the first cohomology of a hyperbolic 3-manifold and the…
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