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By studying connectedness at infinity of systolic groups we distinguish them from some other classes of groups, in particular from the fundamental groups of manifolds covered by euclidean space of dimension at least three. We also study…

We define a natural class of graphs by generalizing prior notions of visibility, allowing the representing regions and sightlines to be arbitrary. We consider mainly the case of compact connected representing regions, proving two results…

We characterize when a generalized Baumslag-Solitar group is linear, and extend the result to the fundamental groups of a graph of groups with infinite virtually cyclic vertex and edge groups.

We consider metric ultraproducts of finite groups with respect to some classes of length functions. All sofic groups embed into these ultraproducts. We study embeddings of normed groups. We also show that in some natural situations such an…

A group is called matricial field (MF) if it admits finite dimensional approximate unitary representations which are approximately faithful and approximately contained in the left regular representation. This paper provides a new class of…

Let M be a compact oriented irreducible 3-manifold which is neither a graph manifold nor a hyperbolic manifold. We prove that the fundamental group of M is virtually special.

A class of models intended to be as minimal and structureless as possible is introduced. Even in cases with simple rules, rich and complex behavior is found to emerge, and striking correspondences to some important core known features of…

Universality has been an important concept in computable structure theory. A class $\mathcal{C}$ of structures is universal if, informally, for any structure, of any kind, there is a structure in $\mathcal{C}$ with the same…

We classify the factorizations of finite classical groups with nonsolvable factors, completing the classification of factorizations of finite almost simple groups.

This is a survey of topological properties of open, complete nonpositively curved manifolds which may have infinite volume. Topics include topology of ends, restrictions on the fundamental group, as well as a review of known examples.

One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry…

Manifolds have uses throughout and beyond Mathematics and it is not surprising that topologists have expended a huge effort in trying to understand them. In this article we are particularly interested in the question: `when is a manifold…

If F is a type-definable family of commensurable subsets, subgroups or sub-vector spaces in a metric structure, then there is an invariant subset, subgroup or sub-vector space commensurable with F. This in particular applies to…

We investigate the mapping class groups of a class of non-Hausdorff topological spaces which includes finite spaces. We show that the mapping class group of a finite space is isomorphic to the homeomorphism group of its $T_0$ quotient. As a…

We prove that a connected, locally finite, quasi-transitive graph which is quasi-isometric to a planar graph is necessarily accessible. This leads to a complete classification of the finitely generated groups which are quasi-isometric to…

In this paper we study obstructions to presentability by products for finitely generated groups. Along the way we develop both the concept of acentral subgroups, and the relations between presentability by products on the one hand, and…

The main subjects of the paper is studying the fundamental groups of closed symplectically aspherical manifolds. Motivated by some results of Gompf, we introduce two classes of fundamental groups $\pi_1(M)$ of symplectically aspherical…

We consider spaces for which there is a notion of harmonicity for complex valued functions defined on them. For instance, this is the case of Riemannian manifolds on one hand, and (metric) graphs on the other hand. We observe that it is…

We show that for every positive integer $n$ there exists a simple group that is of type $\mathrm{F}_{n-1}$ but not of type $\mathrm{F}_n$. For $n\ge 3$ these groups are the first known examples of this kind. They also provide infinitely…

We prove an equivalence of categories from formal complex structures with formal holomorphic maps to homotopy algebras over a simple operad with its associated homotopy morphisms. We extend this equivalence to complex manifolds. A complex…