arXiv research

Algorithm distinguishes Fuchsian groups with finite quotients.

The study shows how quotients of mapping class groups are hierarchically hyperbolic.

Proves Vaisman solvmanifolds are finite quotients of Kodaira-Thurston manifolds.

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.

The study explores finite quotients of 3-manifold groups and their existence and non-existence.

The study distinguishes knots using finite quotients of their fundamental groups.

This paper classifies all admissible quotients of a surface braid group up to order 127.

Finite quotients of fibered hyperbolic 3-manifold groups detect taut polynomials.

The paper studies finiteness of canonical quotients in Dehn quandles of surfaces.

We construct an infinite discrete subgroup of the isometry group of $\mathbb H^3$ 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.

The study characterizes subgroups of braid groups and their finite quotients.

The paper explores mapping class group quotients by Dehn twists and their representations.

New groups from surface braids help create complex geometric shapes.

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.

The study shows a finite number of groups acting on hyperbolic spaces with bounded entropy and compact quotient.

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.

Triangle groups uniquely identified by their finite quotients.

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.

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.

A new presentation of a quotient of braid groups leads to a new type of Burnside group.

Let $W$ be a right-angled Coxeter group corresponding to a finite non-discrete graph $\mathcal{G}$ with at least $3$ vertices. Our main theorem says that $\mathcal{G}^c$ 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.

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.

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 $(2g-2)\times (2g-2)$ matrices over a natural order of $A^{op}$ which preserve a natural skew-Hermitian …

The paper studies binary icosahedral representations of hyperbolic 3-manifolds.

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.

Equivalence proven between divisorial stability and quotient log divisorial stability.

Unified algebraic framework for virtual braid structures with strong structural consequences.

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.

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.

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 $\frac{8}{3}π^{2}$, i.e., they attain all possible volumes of c…

The paper confirms conjectures about Stein manifolds formed by quotients of the ball.

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…

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.