Related papers: Every finite complex has the homology of some CAT(…

In this short note, we show that a group acting geometrically on a CAT(0) cube complex with virtually abelian hyperplane-stabilisers must decompose virtually as a free product of free abelian groups and surface groups.

We construct a model structure on the category of ordered simplicial complexes, Quillen equivalent to the standard model structure on simplicial sets. This shows that simplicial complexes, which are fully combinatorial in nature, provide a…

Quadric complexes are square complexes satisfying a certain combinatorial nonpositive curvature condition. These complexes generalize 2-dimensional CAT(0) cube complexes and are a square analog of systolic complexes. We introduce and study…

This article proposes an algorithm that constructs a Sullivan minimal model for any simply connected simplicial set with effective homology and thereby allows one to decide algorithmically whether two simply connected spaces represented by…

We prove that any group $G$ occurs as $\E(X)$, where $X$ is CW-complex of finite dimension and $\E(X)$ denotes its group of self-homotopy equivalence. Thus, we generalize a well know-theorem due to Costoya and Viruel \cite{CV} asserting…

For an arbitrary separated scheme $X$ of finite type over a finite field $\mathbb F_q$ and an integer $j=-1,-2,$ we prove under the assumption of resolution of singularities, that the two groups $H_{-1}(X,\mathbb Z(j))$ and…

The main technical result of this paper is to characterize the contracting isometries of a CAT(0) cube complex without any assumption on its local finiteness. Afterwards, we introduce the combinatorial boundary of a CAT(0) cube complex, and…

We show that any closed model category of simplicial algebras over an algebraic theory is Quillen equivalent to a proper closed model category. By ``simplicial algebra'' we mean any category of algebras over a simplicial algebraic theory,…

We show that if $Y$ is a compact topological manifold and $X$ is a locally flat submanifold, then the complement $Y - X$ is homotopy equivalent to a finite CW complex. This is a direct proof, and does not rely on much of the theory of…

We construct for every connected locally finite graph $\Pi$ the quantum automorphism group $\text{QAut}\ \Pi$ as a locally compact quantum group. When $\Pi$ is vertex transitive, we associate to $\Pi$ a new unitary tensor category…

Let $X$ be a connected finite CW complex. A connected double covering of $X$ is classified by a non-zero cohomology class $\omega \in H^1(X,\mathbb{Z}_2)$. Denote the double covering space by $X^\omega$. There exists a corresponding…

We construct a functor associating a cubical set to a (simple) graph. We show that cubical sets arising in this way are Kan complexes, and that the A-groups of a graph coincide with the homotopy groups of the associated Kan complex. We use…

Uniformly finite homology is a coarse invariant for metric spaces; in particular, it is a quasi-isometry invariant for finitely generated groups. In this article, we study uniformly finite homology of finitely generated amenable groups and…

We obtain a sufficient condition for lattices in the automorphism group of a finite dimensional CAT(0) cube complex to have infinite girth. As a corollary, we get a version of Girth Alternative for groups acting geometrically: any such…

We prove that a K\"ahler group which is cubulable, i.e. which acts properly discontinuously and cocompactly on a CAT(0) cubical complex, has a finite index subgroup isomorphic to a direct product of surface groups, possibly with a free…

A Rado simplicial complex X is a generalisation of the well-known Rado graph. X is a countable simplicial complex which contains any countable simplicial complex as its induced subcomplex. The Rado simplicial complex is highly symmetric, it…

For any simplicial complex $X$ with a total ordering of its vertices, one can construct a chain complex $\mathbb{L}_\bullet(X)$ generated by necklaces of simplices in $X$, which computes the homology of the free loop space of the geometric…

For a topologically complete space $X$ and a family of closed covers $\mathcal A$ of $X$ satisfying a "local refinement condition" and a "completeness condition," we give a construction of an inverse system $\mathbf{ N}_{\mathcal A}$ of…

We give a sufficient condition for the collapsibility of finite square 2-complexes. We show that any finite, CAT(0) square 2-complex retracts to a point through CAT(0) subspaces.

Let $T$ be a compact torus. We prove that, up to equivariant rational equivalence, the category of $T$-simply connected, $T$-finite type $T$-spaces with finitely many isotropy types is completely described by certain finite systems of…