arXiv research

4-manifolds with specific groups have unique homotopy types.

Constructs Riemannian foliations with exotic tori as leaves.

A myriad of irreducible symplectic 4-manifolds with abelian non-cyclic fundamental group is constructed. The botany of manifolds with finite non-cyclic fundamental groups is also studied.

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.

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

Aspherical manifolds with bounded curvature have non-trivial abelian subgroups in their fundamental groups.

Constructs metrics with positive scalar curvature on odd-order abelian fundamental group manifolds.

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

Study fundamental groups of manifolds with nonnegative Ricci curvature and stability at infinity.

The paper characterizes mapping class groups related to abelian differentials.

The paper proves conditions for self-covering manifolds to be fiber bundles over a circle.

Suppose $X$ and $Y$ are compact connected topological 4-manifolds with fundamental group $π$. For any $r \geqslant 0$, $Y$ is $r$-stably homeomorphic to $X$ if $Y \# r(S^2 \times S^2)$ is homeomorphic to $X \# r(S^2\times S^2)$. How close is stable homeomorphism to homeomorphism? When the common fundamental group $π$ i…

The study shows how nonnegative Ricci curvature and metric cones imply the existence of abelian subgroups in the fundamental group of open manifolds.

Study covers of surfaces, showing types and properties.

We prove that a fundamental group of codimension one nonnegative Ricci curvature C2-foliation of a closed Riemannian manifold is finitely generated and almost abelian, i.e. it contains abelian subgroup of finite index. In particular, we confirm the Milnor conjecture for manifolds which are leaves of codimension one non…

New bounds on Euler characteristics for certain manifolds with finite groups.

We prove that, for any two finite volume hyperbolic $3$-manifolds, the amalgamation of their fundamental groups along any nontrivial geometrically finite subgroup is not LERF. This generalizes the author's previous work on nonLERFness of amalgamations of hyperbolic $3$-manifold groups along abelian subgroups. A consequ…

We show that a complete flat pseudo-Riemannian homogeneous manifold with non-abelian linear holonomy is of dimension at least 14. Due to an example constructed in a previous article by Oliver Baues and the author, this is a sharp bound. Also, we give a structure theory for the fundamental groups of complete flat pseudo…

3-manifolds with two abelian-filled spaces.

New examples show non-abelian fundamental groups for positive Ricci curvature manifolds.

We define cusp-decomposable manifolds and prove smooth rigidity within this class of manifolds. These manifolds generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume, locally symmetric, negatively curved manifolds with cusps. We prove that the gro…

Abelian subgroups in certain spaces are simple.

We show that if $N$, an open connected $n$-manifold with finitely generated fundamental group, is $C^{2}$ foliated by closed planes, then $π_{1}(N)$ is a free group. This implies that if $π_{1}(N)$ has an Abelian subgroup of rank greater than one, then $\mathcal{F}$ has at least a non closed leaf. Next, we show that if…

Let X be a finite CW complex. We show that the fundamental group of X is large if and only if there is a finite cover Y of X and a sequence of finite abelian covers \{Y_N\} of Y which satisfy b_1(Y_N)\geq N. We give some applications of this result to the study of hyperbolic 3--manifolds, mapping classes of surfaces an…

The study finds conditions for nonmaximal topological complexity of manifolds with abelian fundamental groups.

Confirming a conjecture, we show fundamental groups of certain abelian differentials are framed mapping class groups.

The paper shows that certain manifolds with nonnegative Ricci curvature have finitely generated fundamental groups.

The paper characterizes unit balls among Stein spaces with specific groups using Bergman-Einstein metrics.

Proves manifolds with positive scalar curvature have inessential covers.

The geography and botany of smooth/symplectic nonspin 4-manifolds with abelian fundamental group are addressed.

Study finite group actions on Higgs bundle moduli spaces.

The paper connects Artin braid groups to Bieberbach subgroups and flat manifolds with specific holonomy groups.

In this article we study properly discontinuous actions on Hilbert manifolds giving new examples of complete Hilbert manifolds with nonnegative, respectively nonpositive, sectional curvature with infinite fundamental group. We also get examples of complete infinite dimensional Kähler manifolds with positive holomorphic…

The paper proves properties of fundamental groups of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.

Classifies stable diffeomorphism of spin 4-manifolds with specific fundamental groups.

Let $Λ$ be a finite abelian group. A dynamical system with transformation group $Λ$ is a triple $(A,Λ,α)$, consisting of a unital locally convex algebra $A$, the finite abelian group $Λ$ and a group homomorphism $α:Λ\rightarrow\Aut(A)$, which induces an action of $Λ$ on $A$. In this paper we present a new, geometricall…

We show that a closed, connected, oriented, Riemannian $n$-manifold, admitting a branched cover of bounded length distortion from $\mathbb R^n$, has a virtually Abelian fundamental group.

Let $Σ$ be a surface whose interior admits a hyperbolic structure of finite volume. In this paper, we show that any infinite order mapping class acts with infinite order on the homology of some universal $k$--step nilpotent cover of $Σ$. We show that a Torelli mapping class either acts with infinite order on the homolo…

New invariant detects non-homotopy equivalent 4-manifolds.

Study of trigonal curves in abelian differentials with specific divisor properties.

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

We provide an algorithm to solve the word problem in all fundamental groups of closed 3-manifolds; in particular, we show that these groups are autostackable. This provides a common framework for a solution to the word problem in any closed 3-manifold group using finite state automata. We also introduce the notion of a…

We prove that groups that are mod-p-homology equivalent are isomorphic modulo any term of their derived p-series, in precise analogy to Stallings' 1963 result for the lower-central p-series. Similarly spaces that are mod-p-homology equivalent have fundamental groups that are isomorphic modulo any term of their p-derive…

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…

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…

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

We present two results about the relationship between fundamental groups of quasiprojective manifolds and linear systems on a projectivization. We prove the existence of a plane curve with non-abelian fundamental group of the complement which does not admit a mapping onto an orbifold with non-abelian fundamental group.…

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…