arXiv research

3-manifolds with two abelian-filled spaces.

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

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.

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.

4-manifolds with specific groups have unique homotopy types.

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

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

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

Each Abelian subgroup of the fundamental group of a compact and locally simply connected $d$-dimensional length space with no conjugate points is isomorphic to $\mathbb{Z}^k$ for some $0 \leq k \leq d$. It follows from this and previously known results that each solvable subgroup of the fundamental group is a Bieberbac…

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.

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

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

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.

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.…

When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give…

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 paper characterizes mapping class groups related to abelian differentials.

We characterise the virtually abelian groups which are fundamental groups of compact Kähler manifolds and of smooth projective varieties. We show that a virtually abelian group is Kähler if and only if it is projective. In particular, this allows to describe the Kähler condition for such groups in terms of integral sym…

For certain manifolds, nonnegative Ricci curvature limits dimension and forces almost abelian fundamental group.

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

We construct smooth fiber bundles such that the fibers are exotic tori and the total space has finite abelian fundamental group. This gives examples of a Riemannian foliation on a closed manifold whose leaves are exotic tori and whose total space has finite abelian fundamental group.

A 5-manifold fibers over a circle with nonpositive curvature.

Determines conditions for abelian differentials with specific singularities.

The geography of minimal symplectic 4-manifolds with arbitrary fundamental group and symplectic 6-manifolds with abelian fundamental group of small rank, and with arbitrary fundamental group are addressed.

We introduce Riemannian metrics of positive scalar curvature on manifolds with Baas-Sullivan singularities, prove a corresponding homology invariance principle and discuss admissible products. Using this theory we construct positive scalar curvature metrics on closed smooth manifolds of dimension at least five which ha…

Fukaya-Yamaguchi conjecture holds in 4D manifolds with nonnegative curvature.

Study of complex and Hermitian structures on specific Lie groups.

Study covers of surfaces, showing types and properties.

Study on spheres in simply-connected 4-manifolds with abelian complements.

We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and non-complete examples, where the linear ho…

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…

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…

A theorem simplifies mass-minimizing flat chains' regularity.

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

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

A square complex is a 2-complex formed by gluing squares together. This article is concerned with the fundamental group $Γ$ of certain square complexes of nonpositive curvature, related to quaternion algebras. The abelian subgroup structure of $Γ$ is studied in some detail.

Geodesic loops escape from balls at a sublinear rate imply virtually abelian fundamental group.

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…

Anabelian geometry reformulated using Hodge theory for hyperbolic curves.

New smooth structures found on certain 4D spaces.

Classifies Spin(7) structures on compact 8-manifolds with abelian fundamental group.

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

Open manifolds with nonnegative Ricci curvature have virtually abelian fundamental groups if they escape from bounded balls at a small rate.

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…

A group morphism is constructed, which can be realized as the induced morphism of fundamental groups from a holomorphic map between compact Kahler manifolds, but can not be realized by a holomorphic map between smooth projective varieties. And it is also proved that there exists no such example between abelian groups.

New group-theoretic Johnson classes applied to curves with torsion Ceresa classes.

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…