arXiv research

Study of uncountable family of finitely generated groups.

Finite type and finitely generated homotopy groups for manifold automorphisms.

The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group $G$, Kapovich provided a partial algorithm which, on input a finite set $S$ of $G$, halts if $S$ generates a quasiconvex subgroup of $G$ and run…

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is resudually finite. We…

The paper characterizes Alexander quandles of finite groups.

In geometric group theory one uses group actions on spaces to gain information about groups. One natural space to use is the Cayley graph of a group. The Cayley graph arguments that one encounters tend to require local finiteness, and hence finite generation of the group. In this paper, I take the theory of intersectio…

New computations show various properties of bounded cohomology in finitely presented groups.

The study proves how groups can be split with limited complexity.

We construct analogues of FI-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity. Applications include a proof of the Lannes--Schwartz Artinian conjecture in the generic representatio…

A generalized Baumslag-Solitar (GBS) group is a finitely generated group acting on a tree with infinite cyclic edge and vertex stabilizers. We show how to determine effectively the rank (minimal cardinality of a generating set) of a GBS group; as a consequence, one can compute the rank of the mapping torus of a finite …

We give a simple proof of the finite presentation of Sela's limit groups by using free actions on R^n-trees. We first prove that Sela's limit groups do have a free action on an R^n-tree. We then prove that a finitely generated group having a free action on an R^n-tree can be obtained from free abelian groups and surfac…

Groups with semistable peripheral subgroups are semistable.

3-manifold groups are uniquely identifiable via their profinite completions.

We give an exposition of Delzant's ideas extending the notion of Scott complexity of finitely generated groups to surjective homomorphisms of finitely presented groups to finitely generated groups.

Torelli group's second homology is finite for large genus surfaces.

We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pur…

In this paper we prove trace formulae for the Reidemeister number of a group endomorphism. This result implies the rationality of the Reidemeister zeta function in the following cases: the group is a direct product of a finite group and a finitely generated Abelian group; the group is finitely generated, nilpotent and …

Paper proves Torelli group's finiteness for surfaces with 2 boundaries.

Triangle groups uniquely identified by their finite quotients.

Study covers of Chamanara surface with large Veech groups.

In this paper we provide a criteria for geometric finiteness of Kleinian groups in general dimension. We formulate the concept of conformal finiteness for Kleinian groups in space of dimension higher than two, which generalizes the notion of analytic finiteness in dimension two. Then we extend the argument in the paper…

Many 2D Artin groups are residually finite.

Finite index subgroups of relatively hyperbolic groups have equal index.

The study characterizes subgroups of mapping tori of free groups.

The study examines Hopfian properties of conjugation quandles and their underlying groups.

Flawed groups are shown to include all finitely generated groups isomorphic to free products of nilpotent groups.

Genus 3 Heegaard groups of lens space connected sums are finitely generated.

Study SL(2,C) character schemes for finitely generated groups.

We show that every graph product of finitely generated abelian groups acts properly and cocompactly on a CAT(0) cubical complex. The complex generalizes (up to subdivision) the Salvetti complex of a right-angled Artin group and the Coxeter complex of a right-angled Coxeter group. In the right-angled Artin group case it…

This thesis introduces big mapping class groups and their structure.

We survey the existing parts of a classification of finite groups generated by orthogonal transformations in a finite-dimensional Euclidean space whose fixed point subspace has codimension one or two and extend it to a complete classification. These groups naturally arise in the study of the quotient of a Euclidean spa…

Groups with similar pro-$p$ completions have two-generated subgroups that are free.

Paper shows 3D hyperbolic manifolds are uniquely identified by their finite groups.

Extensions of Veech groups using hierarchical hyperbolic spaces.

It is proven that if a finitely presented group is one ended it has asymptotic dimension bigger than one. It follows that finitely presented groups with asdim 1 are virtually free. A counterexample is given for the finitely generated case.

Open manifolds with nonnegative Ricci curvature and Euclidean volume growth have finitely generated fundamental groups.

Study of generalized J-groups and their presentations.

The paper studies groups with specific actions on hyperbolic spaces and finds that subgroups are either amenable or contain a free group.

Investigates BNSR invariants of link and knot groups, proving specific properties.

We show that any Kahler extension of a finitely generated abelian group by a surface group of genus g at least 2 is virtually a product. Conversely, we prove that any homomorphism of an even rank, finitely generated abelian group into the genus g mapping class group with finite image gives rise to a Kahler extension. T…

We construct a finitely generated group that does not satisfy the generalized Burghelea conjecture.

4-manifolds with specific groups have unique homotopy types.

The paper constructs finite generating sets for complex algebraic structures.

Paper compares two entropy concepts for finite presentation groups.

Given a prime $p$, a group is called residually $p$ if the intersection of its $p$-power index normal subgroups is trivial. A group is called virtually residually $p$ if it has a finite index subgroup which is residually $p$. It is well-known that finitely generated linear groups over fields of characteristic zero are …

We present a method for computing the number of epimorphisms from a finitely-presented group G to a finite solvable group Γ, which generalizes a formula of Gäschutz. Key to this approach are the degree 1 and 2 cohomology groups of G, with certain twisted coefficients. As an application, we count low-index subgroups of …

Investigates linearity of group amalgams and new examples of non-linear groups.

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…