arXiv research

The paper classifies fixed subgroups of endomorphisms in free-abelian times surface groups.

The isomorphism problem for [free abelian]-by-free groups is unsolvable.

The study shows that certain groups can be uniquely identified by their finite abelian summands.

We classify up to coarse equivalence all countable abelian groups of finite torsion free rank. The Q-cohomological dimension and the torsion free rank are the two invariants that give us such classification. We also prove that any countable abelian group of finite torsion free rank is coarsely equivalent to Z^n + H whe…

It is known that if a classical link group is a free abelian group, then its rank is at most two. It is also known that a $k$-component 2-link group ($k>1$) is not free abelian. In this paper, we give examples of $T^2$-links each of whose link groups is a free abelian group of rank three or four. Concerning the $T^2$-l…

Suppose a group $G$ is quasi-isometric to a free product of a finite set $S$ of finitely generated abelian groups; let $S'$ denote the set of ranks of the free abelian parts of the groups in $S$. Then $G$ is commensurable with the free product of $\Z$ with a $\Z^n$ for each $n$ occurring in $S'$.

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…

We discuss the possibility of lifting finite subgroups, and in particular finite cyclic subgroups, with respect to the canonical projections between automorphism and outer automorphism groups of free groups, surface groups and their abelianizations.

The study finds a subgroup of graph braid groups that is a direct product of non-abelian free groups.

New proof and description of commutator subgroups for free and surface groups.

We show, using Wise's equitable sets criterion, that every tubular free by cyclic group acts freely on a CAT(0) cube complex. We also show that these groups have a finite index subgroup satisfying the strongest Tits alternative, which means that every subgroup either surjects a non abelian free group or is torsion free…

Study on spectral sequence for abelian Lie group actions, with bounds and applications.

The fundamental group of the complement of a plane curve is a very important topological invariant. In particular, it is interesting to find out whether this group is determined by the combinatorics of the curve or not, and whether it is a direct sum of free groups and a free abelian group, or it has a conjugation-free…

The Hausmann-Weinberger invariant of a group G is the minimal Euler characteristic of a closed orientable 4-manifold M with fundamental group G. We compute this invariant for finitely generated free abelian groups and estimate the invariant for all finitely generated abelian groups.

Study covers of Chamanara surface with large Veech groups.

The paper constructs a free abelian group from Anosov representations on bundles.

The paper finds small exotic 4-manifolds with free abelian groups.

In this paper we give an example of a linear group such that its tensor square is not linear. Also, we formulate some sufficient conditions for the linearity of non-abelian tensor products $G \otimes H$ and tensor squares $G \otimes G$. Using these results we prove that tensor squares of some groups with one relation a…

We study surface links whose link groups are free abelian, and construct various stimulating and highly non-trivial examples of such surface links.

Characterizes almost Abelian Lie algebras with special $H$-structures.

We present a method for deciding when a regular abelian cover of a finite CW-complex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finite CW-complex X, with group of deck transformations a fixed abelian group A, which in the case of free abelian covers of ran…

Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. Assuming certain conditions on the fundamental group $π_1(M)$ we construct quasi-isometric embeddings of either free Abelian or direct products of non-Abelian free groups into the group of volume preserving diffe…

We define an integer-valued invariant of special cube complexes called the genus, and prove that having genus one characterizes special cube complexes with abelian fundamental group. Using the genus, we obtain a new proof that the fundamental group of a special cube complex is either free abelian or surjects onto a non…

Study shows non-abelian free groups' 4th cohomology is non-zero.

We give a description of the model theoretic relation of forking independence in terms of the notion of JSJ decompositions in non abelian free groups.

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 …

We establish a close connection between stable commutator length in free groups and the geometry of sails (roughly, the boundary of the convex hull of the set of integer lattice points) in integral polyhedral cones. This connection allows us to show that the scl norm is piecewise rational linear in free products of Abe…

The study extends group actions from surfaces to 3-manifolds using $G$-cobordisms.

Birman-Lubotzky-McCarthy proved that any abelian subgroup of the mapping class groups for orientable surfaces is finitely generated. We apply Birman-Lubotzky-McCarthy's arguments to the mapping class groups for non-orientable surfaces. We especially find a finitely generated group isomorphic to a given torsion-free sub…

Study shows complex K3 surfaces have infinite free abelian subgroup in their diffeomorphism group.

We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and …

We present an explicit realization of abelian extensions of infinite dimensional Lie groups using abelian extensions of path groups, by generalizing Mickelsson's approach to loop groups and the approach of Losev-Moore-Nekrasov-Shatashvili to current groups. We apply our method to coupled cocycles on current Lie algebra…

We consider linear groups which do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic, and show that this class of groups has good properties which resemble those held by groups of non positive curvature and which do not hold for arbitrary characteristic zero l…

Study shows compact mapping class groups of infinite type surfaces are never perfect.

Trivial Massey product in specific cohomology groups.

We show that if a group is not virtually cyclic and is hyperbolic relative to a family of proper subgroups, then it has a hyperbolically embedded subgroup which contains a finitely generated non-abelian free group as a finite index subgroup.

Researchers show how to perturb free group representations into higher rank groups.

The study explores automorphisms and centralizers in free group outer automorphisms.

Recall that a group is called large if it has a finite index subgroup which surjects onto a non-abelian free group. By work of Agol and Cooper-Long-Reid, most 3-manifold groups are large; in particular, the fundamental groups of hyperbolic 3-manifolds are large. In previous work, the first author gave examples of close…

Closed self-covering manifolds with abelian fundamental groups fiber over tori.

A generalized Baumslag-Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains non-abelian free groups or is virtually nilpotent of class at most 2. It has torsion only at finitely many primes. One may dec…

We prove that, except for a few explicit roots of unity, the quantum image of any Johnson subgroup of the mapping class group contains an explicit free non-abelian subgroup.

Maps between automorphism groups are isomorphisms for free factor complexes.

We study the large scale geometry of the mapping class group, MCG. Our main result is that for any asymptotic cone of MCG, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG. An application is an affirmative solution to Brock-Farb's Rank Conjecture which as…

The fundamental group of the complement of a hyperplane arrangement plays an important role in studying the corresponding arrangements. In particular, for large families of hyperplane arrangements, this fundamental group, being isomorphic to the fundamental group of a complement of a line arrangement, has some remarkab…

We show that stable commutator length is rational on free products of free Abelian groups amalgamated over $\mathbb{Z}^k$, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parameterize all surfaces with specified boundary mapping to th…

Probabilistic proof shows separability in free groups.

Homotopy braid groups are shown to be torsion-free.