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.
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.
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 study explores finite quotients of 3-manifold groups and their existence and non-existence.
problem Does there exist a 3-manifold group with a specific finite quotient but not others?
method The approach combines group cohomology, topological results, and probabilistic methods.
result Proves existence and non-existence of 3-manifolds with certain finite quotients.
We show that for any positive integer m≥1, m-relator quotients of the modular group M=PSL(2,Z) generically satisfy a very strong Mostow-type \emph{isomorphism rigidity}. We also prove that such quotients are generically "essentially incompressible". By this we mean that their "absolute T-invariant…
Study ratio-limit boundaries for random walks on hyperbolic groups.
problem Computing ratio-limit boundaries for relatively hyperbolic groups.
method Adapting Woess's strategy to non-hyperbolic groups and analyzing degenerate cases.
result Closure of minimal points in R-Martin boundary is the unique smallest invariant subspace in ratio-limit boundary. We study the ends of a generic manifold, with respect to a unimodular measure on the space of pointed Riemannian manifolds with bounded curvatures. We apply our general result to the case of surfaces and obtain as corollaries a very precise description of generic leaves for foliations with invariant measures and of quo…
Random groups with high density have Property (T).
problem Determining the conditions for Property (T) in random groups.
method Analyzing k-gonal random groups and their limits. result Property (T) holds with high probability for high densities.
A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficul…
We introduce the notion of a stationary random manifold and develop the basic entropy theory for it. Examples include manifolds admitting a compact quotient under isometries and generic leaves of a compact foliation. We prove that the entropy of an ergodic stationary random manifold is zero if and only if the manifold …
We introduce a new random group model called the square model: we quotient a free group on n generators by a random set of relations, each of which is a reduced word of length four. We prove, as in the Gromov density model, that for densities >21 a random group in the square model is trivial with overwhel…
Sharp-SSL uses random projections to identify important variables for semi-supervised learning.
problem High-dimensional semi-supervised learning problems.
method Careful aggregation of low-dimensional results from many axis-aligned random projections.
result Sharp-SSL algorithm can recover signal coordinates with high probability.
We consider two random group models: the hexagonal model and the square model, defined as the quotient of a free group by a random set of reduced words of length four and six respectively. Our first main result is that in this model there exists a sharp density threshold for Kazhdan's Property (T) and it equals 1/3. Ou…
The quotient of random variables with normal distributions is examined and proven to have have power law decay, with density f(x)≃f0x−2, with the coefficient depending on the means and variances of the numerator and denominator and their correlation. We also obtain the conditional probability…
The study connects Kleinian group divergence to random walk recurrence.
problem Understanding the recurrence of random walks on Schreier graphs of Kleinian groups.
method Connecting growth rates of orbits, volume, and Schreier graphs.
result Constructing Kleinian groups of divergence type.
We study the problem of forecasting volatility for the multifractal random walk model. In order to avoid the ill posed problem of estimating the correlation length T of the model, we introduce a limiting object defined in a quotient space; formally, this object is an infinite range logvolatility. For this object and th…
We prove sharp limit theorems on random walks on graphs with values in finite groups. We then apply these results (together with some elementary algebraic geometry, number theory, and representation theory) to finite quotients of lattices in semisimple Lie groups (specifically SL(n,Z) and Sp(2n, Z) to show that a ``ran…
We prove a generalization of the fundamental inequality of Guivarc'h relating entropy, drift and critical exponent to Gibbs measures on geometrically finite quotients of CAT(-1) metric spaces. For random walks with finite superexponential moment, we show that the equality is achieved if and only if the Gibbs density is…
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.
Sharp bounds found on nonabelian quotients of surface braid groups.
problem Finding the smallest nonabelian quotients of surface braid groups.
method Sharp lower bounds and classification of quotients.
result Quotients of minimum order are either symmetric groups or 2-step nilpotent p-groups.
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 finds smallest non-trivial quotients of braid groups and commutator subgroups.
problem Identifying smallest non-trivial quotients of braid groups and commutator subgroups.
method Analyzing symmetric and alternating groups for quotients of braid groups and commutator subgroups.
result Proved smallest quotients for specific braid groups and commutator subgroups.
In this paper, we propose a weak version of quotient for the algebraic action of a group on a variety, which we shall call a pseudo-quotient. They arise when we focus on the purely topological properties of good GIT quotients regardless of their algebraic properties. The flexibility granted by their topological nature …
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…
Paper compares total quotient curvature and proves bounds for Einstein metric.
problem Comparing total quotient curvature and proving bounds for Einstein metric.
method Established comparison theorems and proved integral inequalities.
result Background Einstein metric achieves a sharp bound on total quotient curvature.
Establishes a concavity property for positive Hessian quotient operators.
problem Analyzing positive Hessian quotient operators on Riemannian manifolds.
method Proves a special concavity property and a Jacobi inequality.
result Proves a Jacobi inequality for symmetric tensors.
Paper examines properties of specific solutions to Yamabe flow.
problem Characterizing quotient almost Yamabe solitons.
method Investigates quotient almost Yamabe solitons and presents conditions for their rigidity.
result Sufficient conditions for quotient almost Yamabe solitons to be trivial or isometric with a sphere.
We study Deraux's non arithmetic orbifold ball quotient surfaces obtained as birational transformations of a quotient X of a particular Abelian surface A. Using the fact that A is the Jacobian of the Bolza genus 2 curve, we identify X as the weighted projective plane P(1,3,8). We compute the equati…
Study shows quotient group is infinitely generated.
problem Understanding the structure of homology cobordism groups.
method Proved using Seifert fibered spaces.
result Quotient group is infinitely generated.
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.
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 on Kohn Laplacian spectrum on sphere quotients.
problem Determining fundamental group from Kohn Laplacian spectrum.
method Weyl-type theorem, CR manifolds, Sobolev estimates.
result Fundamental group can be determined from Kohn Laplacian spectrum in 3D.
Algorithm distinguishes Fuchsian groups with finite quotients.
problem Distinguishing between non-isomorphic Fuchsian groups.
method Develops an algorithm to create group extensions using finite quotients.
result Establishes an upperbound for the order of a distinguishing finite quotient.
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.
The paper studies Lie algebroid and groupoid quotients with forms, applying to Poisson and Dirac structures.
problem Understanding quotients of Lie algebroids and groupoids with compatible differential forms.
method Identifying Lie theoretic conditions for forms to be basic, characterizing induced forms on quotients, and applying results to Poisson and Dirac structures.
result Recovery and generalization of known results on Poisson reduction.
Reductive quotients preserve klt singularities in algebraic geometry.
problem Preserving klt singularities in quotients of klt singularities.
method Proving that the quotient of a klt type singularity by a reductive group is of klt type.
result The quotient of a klt variety by a reductive group results in a klt variety with a suitable boundary.
The center of a quotient group of piecewise linear homeomorphisms is trivial.
problem Understanding the structure of a specific group of homeomorphisms.
method Analyzing a quotient of piecewise linear homeomorphisms of the real line.
result The center of the quotient group is trivial.
The study provides interior curvature estimates for convex graphs satisfying a specific quotient equation.
problem Interior curvature estimates for convex graphs.
method Analyzes convex graphs satisfying the quotient equation σn−2σn(λ)=f(X)>0. result Interior curvature estimates for convex graphs.
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.
Short note finds a new metric from sphere quotients.
problem Finding metrics from sphere quotients.
method Conformal stereographic projection on sphere quotients.
result Result is a Majumdar-Papapetrou metric.
Study non-existence of complex ball quotients in Torelli locus.
problem Non-existence of totally geodesic complex ball quotients in Torelli locus.
method Analytic techniques.
result Analytic techniques used to study non-existence.
Paper studies Hessian quotient equations in warped product manifolds.
problem Analyzing Hessian quotient equations in warped product manifolds.
method Using standard degree theory and a priori estimates.
result Existence of star-shaped compact hypersurface solutions.
Study proves estimate for Hessian quotient equations on 2D Riemannian manifolds.
problem Problems posed by Delanoë and Urbas related to Hessian quotient equations.
method Maximum principle argument and new test function introduced to prove estimate.
result Unobstructed second order a priori estimate for real Hessian quotient equation in 2D.
Estimates solutions to Hessian quotient equations on HKT manifolds.
problem Finding C0 estimates for solutions to Hessian quotient equations. method Using the cone condition directly to show C0 estimates. result Showed C0 estimates for solutions to Hessian quotient equations on HKT manifolds. We study a quotient of the group algebra of the braid group in which the Artin generators satisfy a cubic relation. This quotient is maximal among the ones satisfying such a cubic relation. It is finite-dimensional for at least n at most 5 and we investigate its module structure in this range. We also investigate the p…
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…
Study on braid groups' congruence subgroups and their crystallographic quotients.
problem Understanding congruence subgroups and crystallographic quotients of braid groups.
method Investigation of lower central series of congruence braid groups related to B3. result Quotients of congruence braid groups are almost crystallographic.
This paper classifies all admissible quotients of a surface braid group up to order 127.
problem Classifying admissible quotients of surface braid groups with bounded order.
method Analyzing finite quotients of the pure braid group on two strands of a Riemann surface.
result All admissible quotients of the braid group have order at most 127.