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 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.
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.
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 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.
New proof shows smallest non-cyclic automorphism group quotients are linear 2-groups.
problem Identifying the smallest non-cyclic quotients of free group automorphism groups.
method Elementary proof using standard projection and automorphisms.
result All minimal quotients are linear groups over the field of two elements.
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.
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.
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.
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.
Paper explores relations between braid groups and their quotients.
problem Understanding relations between braid groups and their quotients.
method Recalling and introducing elements of congruence braid groups, establishing isomorphisms between crystallographic and congruence braid groups.
result Established isomorphisms between crystallographic braid groups and quotients of congruence braid groups.
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 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.
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.
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 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.
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.
The paper studies Coxeter quotients of surface braid groups.
problem Analyzing Coxeter quotients of surface braid groups.
method Analyzing Coxeter-type quotients of surface braid groups by normal closure and commutator relations.
result Characterized Coxeter quotients of surface braid groups.
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.
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. 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.
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.
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.
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.
The study explores congruence subgroups of braid groups and their quotients.
problem Understanding the structure of congruence subgroups of braid groups.
method Utilizing the integral Burau representation and results from integral matrices, the study examines quotients of these subgroups.
result Findings of quotients that are not isomorphic to symmetric groups.
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.
Sharp growth tightness proven for group quotients.
problem Growth behavior of group quotients by confined subgroups.
method Statistically convex-cocompact action with contracting elements.
result Sharp growth tightness proven, with applications to uniformly recurrent subgroups.
No exact G₂-structures on compact Lie group quotients.
problem Existence of exact G₂-structures on compact quotients of Lie groups.
method Analyzing compact quotients of seven-dimensional Lie groups by co-compact discrete subgroups.
result Compact quotients of seven-dimensional Lie groups by co-compact discrete subgroups do not admit exact G₂-structures induced by left-invariant ones.
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.
We derive a lower bound on the size of finite non-cyclic quotients of the braid group that is superexponential in the number of strands. We also derive a similar lower bound for nontrivial finite quotients of the commutator subgroup of the braid group.
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 …
New groups from surface braids help create complex geometric shapes.
problem Creating new geometric shapes from surface braids.
method Using finite quotients of surface braid groups to construct double Kodaira fibrations.
result New geometric shapes (double Kodaira fibrations) created using these groups.
We construct an infinite discrete subgroup of the isometry group of H3 with no finite quotients other than the trivial group.
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…
Three versions of the Freiheitssatz are proved in the context of one-relator quotients of limit groups, where the latter are equipped with 1-acylindrical splittings over cyclic subgroups. These are natural extensions of previously published corresponding statements for one-relator quotients of orientable surface groups…
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 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.
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 groups defined from knot diagrams, invariant under Reidemeister moves.
problem Classical knot groups are not invariant under all Reidemeister moves.
method Define quotient groups based on knot diagrams, invariant under Reidemeister moves.
result New groups include extended knot groups and are invariant under all Reidemeister moves.
Maximal rank Coxeter quotients found for 1.7M knots up to 16 crossings.
problem Finding maximal rank Coxeter quotients for knots.
method Computational approach to find Coxeter quotients for knots up to 16 crossings.
result Verification of Meridional Rank Conjecture for 595,515 knots.
New manifold types defined on quotient spaces.
problem Characterizing specific manifold structures.
method Introducing decomposable LCP manifolds and studying their properties.
result Characterized LCP manifolds on quotient spaces.
Proves Vaisman solvmanifolds are finite quotients of Kodaira-Thurston manifolds.
problem Characterizing Vaisman solvmanifolds and their properties.
method Analyzing fundamental groups and quotient structures.
result Every Vaisman solvmanifold is a finite quotient of a Kodaira-Thurston manifold.
The paper explores conditions for topological rigidity in quotients of the Davis complex.
problem Understanding when quotients of the Davis complex are topologically rigid.
method Analyzing quotients of the Davis complex of right-angled Coxeter groups and conditions on defining graphs.
result Introduction of infinitely many infinite topologically rigid subclasses.
The study characterizes elements of a group quotient and finds a non-locally indicable left-orderable group.
problem Understanding the structure and properties of a specific group quotient.
method Introduced an invariant for quasi-isometries of the positive real line and split it into units.
result Found a quotient of the quasi-isometry group of the positive real line that is left-orderable but not locally indicable.
We generalize the hyperkaehler quotient construction to the situation where there is no group action preserving the hyperkaehler structure but for each complex structure there is an action of a complex group preserving the corresponding complex symplectic structure. Many (known and new) hyperkaehler manifolds arise as …
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.
We determine all generalized Baumslag-Solitar groups (finitely generated groups acting on a tree with all stabilizers infinite cyclic) which are quotients of a given Baumslag-Solitar group BS(m,n), and (when BS(m,n) is not Hopfian) which of them also admit BS(m,n) as a quotient. We determine for which values of r,s one…
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.