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.
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.
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.
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.
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 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 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.
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.
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
The paper studies finiteness of canonical quotients in Dehn quandles of surfaces.
problem Finiteness of canonical quotients in Dehn quandles of surfaces.
method Analyzing the Dehn quandle structure and using quandle quotients.
result For surfaces of positive genus, the n-quandle of their Dehn quandles is finite for specific values of n. We construct an infinite discrete subgroup of the isometry group of H3 with no finite quotients other than the trivial group.
In this paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying the theory of Variation of Geometric Invariant Theory Quotients ([3]), we show tha…
Classifies links with finite N-quandles for some N.
problem Describing finite quotients of quandles, especially n-quandles. method Introducing and studying N-quandles, proving one direction of a conjectured classification. result Proves one direction of a conjectured classification of links with finite N-quandles. The study characterizes subgroups of braid groups and their finite quotients.
problem Understanding the relationship between braid groups and their congruence subgroups.
method Characterization and computation of finite quotients of braid groups.
result Explicit generators and free subgroups are found for certain braid 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.
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 characterize finite groups G generated by orthogonal transformations in a finite-dimensional Euclidean space V whose fixed point subspace has codimension one or two in terms of the corresponding quotient space V/G with its quotient piecewise linear structure.
The paper characterizes unit balls among Stein spaces with specific groups using Bergman-Einstein metrics.
problem Characterizing unit balls among Stein spaces with specific groups.
method Study of Bergman metric on finite ball quotients and its Kähler-Einstein property.
result The Bergman-Einstein metric exists only on the unit ball itself for finite ball quotients with trivial groups.
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 study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.
We show that there is no algorithm deciding whether the maximal residually free quotient of a given finitely presented group is finitely presentable or not. Given a finitely generated subgroup G of a finite product of limit groups, we discuss the possibility of finding an explicit set of defining equations (i.e. of exp…
We give examples of closed, oriented 3-manifolds whose fundamental groups are not isomorphic, but yet have the same sets of finite quotient groups; hence the same profinite completions. We also give examples of compact, oriented 3-manifolds with non-empty boundaries whose fundamental groups though isomorphic have disti…
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.
Triangle groups uniquely identified by their finite quotients.
problem Identifying triangle groups among finitely generated residually finite groups.
method Character varieties method to distinguish profinite completions.
result Certain Fuchsian triangle groups are profinitely rigid.
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.
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.
We state and prove a condition under which the strong Atiyah Conjecture carries over to subgroups. Moreover, we show that if a group satisfies the (strong) Atiyah Conjecture then any quotient with finite kernel does.
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.
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.
Let W be a right-angled Coxeter group corresponding to a finite non-discrete graph G with at least 3 vertices. Our main theorem says that Gc is connected if and only if for any infinite index quasiconvex subgroup H of W and any finite subset $\{ γ_1, \ldots , γ_n \} \subset W \setminus …
The article characterizes complex torus quotients with numerical conditions.
problem Characterizing quotients of complex tori by finite groups.
method Numerical vanishing condition on Chern classes, Bogomolov-Gieseker inequality for singular spaces.
result Generalization of previous results in projective and three-dimensional settings.
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…
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.
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 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.
We describe a pair of invariants for actions of finite groups on shifts of finite type, the left-reduced and right-reduced shifts. The left-reduced shift was first constructed by U. Fiebig, who showed that its zeta function is an invariant, and in fact equal to the zeta function of the quotient dynamical system. We als…
A well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a once-punctured torus bundle over the circle. We give substantial compu…
The study of infinite groups through their finite quotients in geometry.
problem Understanding properties of infinite groups from their finite images.
method Analyzing infinite groups through their finite quotients and using low-dimensional topology.
result Recent results show how finite images can determine the group completely in some cases.
Unified algebraic framework for virtual braid structures with strong structural consequences.
problem Unified algebraic framework for virtual braid structures with various types of crossings.
method Introducing the universal virtual braid group UVn(c) and proving its properties. result Strong structural consequences including residual finiteness, linearity, and solvability of conjugacy problems.
Equivalence proven between divisorial stability and quotient log divisorial stability.
problem Equivalence of divisorial stability and log divisorial stability under finite group actions.
method Interpolation technique and equivariant divisorial stability construction.
result Equivariant divisorial stability of a polarized variety is equivalent to log divisorial stability of its quotient.
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.
Study shows Bergman kernels match averages on quotient spaces, proving non-vanishing of Poincaré series.
problem Proving non-vanishing of Poincaré series on finite-volume quotients of Hermitian symmetric spaces.
method Using Bergman kernels and averaging over discrete groups, proving non-vanishing of Poincaré series.
result Large class of relative Poincaré series does not vanish on general locally symmetric spaces of finite volume.
The paper studies a degenerate equation related to Kähler-Ricci flow on symplectic quotients.
problem Finite time singularities of the Kähler-Ricci flow on symplectic quotients.
method Interpreting the V-soliton equation and reducing it to a scalar equation on Kähler potentials. result Preliminary estimates for the scalar equation on compact Kähler manifolds.
A Coxeter group acts properly and cocompactly by isometries on the Davis complex for the group; we call the quotient of the Davis complex under this action the Davis orbicomplex for the group. We prove the set of finite covers of the Davis orbicomplexes for the set of one-ended Coxeter groups is not topologically rigid…
Study extra-special quotients of surface braid groups and construct double Kodaira fibrations.
problem Investigate non-abelian finite quotients of surface braid groups and double Kodaira fibrations with small signature.
method Introduced diagonal double Kodaira structures to study finite quotients of pure braid groups and constructed double Kodaira fibrations.
result Proved that if a finite group admits a diagonal double Kodaira structure, then its order is at least 32, with equality if and only if the group is extra-special.
We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of 38π2, i.e., they attain all possible volumes of c…
The paper confirms conjectures about Stein manifolds formed by quotients of the ball.
problem Characterizing Stein manifolds formed by quotients of the ball.
method Analyzing discrete subgroups of PU(n,1) and their quotients.
result The quotient of the ball by geometrically finite groups is Stein.
We construct some non-arithmetic ball quotients as branched covers of a quotient of an Abelian surface by a finite group, and compare them with lattices that previously appear in the literature. This gives an alternative construction, which is independent of the computer, of some lattices constructed by the author with…