Related papers: A link condition for simplicial complexes, and CUB…

A simplicial complex is called negatively curved if all its simplices are isometric to simplices in hyperbolic space, and it satisfies Gromov's Link Condition. We prove that, subject to certain conditions, a compact graph of spaces whose…

We provide a condition on the links of polygonal complexes that is sufficient to ensure groups acting properly discontinuously and cocompactly on such complexes contain a virtually free codimension-1 subgroup. We provide stronger conditions…

We provide a necessary and sufficient condition on a finite flag simplicial complex, L, for which there exists a unique CAT(0) cube complex whose vertex links are all isomorphic to L. We then find new examples of such CAT(0) cube complexes…

For a CAT(0) cube complex $\mathbf X$, we define a simplicial flag complex $\partial_\Delta\mathbf X$, called the \emph{simplicial boundary}, which is a natural setting for studying non-hyperbolic behavior of $\mathbf X$. We compare…

The purpose of this article is to study directed collapsibility of directed Euclidean cubical complexes. One application of this is in the nontrivial task of verifying the execution of concurrent programs. The classical definition of…

In this article we study combinatorial non-positive curvature aspects of various simplicial complexes with natural $\widetilde A_n$ shaped simplicies, including Euclidean buildings of type $\widetilde A_n$ and Cayley graphs of Garside…

We introduce and investigate bucolic complexes, a common generalization of systolic complexes and of CAT(0) cubical complexes. They are defined as simply connected prism complexes satisfying some local combinatorial conditions. We study…

We explore a local combinatorial condition on a simplicial complex, called 8-location. We show that 8-location and simple connectivity imply Gromov hyperbolicity. We mention some applications of such combinatorial negative curvature…

In this article, we are interested in metric spaces that satisfy a weak non-positive curvature condition in the sense that they admit a conical geodesic bicombing. We show that the analog of a question of Gromov about compactness properties…

We consider finite 2-dimensional polyhedral complexes, equipped with piecewise non-positively curved, locally CAT(0) metrics. We give conditions on the complex X that ensure that its fundamental group contains a surface subgroup. Concrete…

We introduce the notion of coupled link cube complex (CLCC) as a means of constructing interesting cocompactly cubulated groups. CLCCs are defined locally, making them a useful tool when precise control over the links is required. In this…

We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$, if for every $\epsilon > 0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less…

We introduce the polygonalisation complex of a surface, a cube complex whose vertices correspond to polygonalisations. This is a geometric model for the mapping class group and it is motivated by works of Harer, Mosher and Penner. Using…

Let G be the group of rational points of a semisimple algebraic group of rank 1 over a nonarchimedean local field. We improve upon Lubotzky's analysis of graphs of groups describing the action of lattices in G on its Bruhat-Tits tree…

We give a proof to the following theorem, which is well-known among experts: A connected subcomplex $W$ of a finite dimensional CAT(0) cubed complex $X$ is convex if and only if Lk$(v, W)$ is a full subcomplex of Lk$(v, X)$ for every vertex…

We consider closed simplicial and cubical $n$-complexes in terms of link of their $(n-2)$-faces. Especially, we consider the case, when this link has size 3 or 4, i.e., every $(n-2)$-face is contained in 3 or 4 $n$-faces. Such simplicial…

We examine a condition on a simply connected 2-complex X ensuring that groups acting properly on X are coherent. This extends earlier work on 2-complexes with negative sectional curvature which covers the case that G acts freely. Our…

In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales. We show that some such results remain valid for metric spaces with non-unique…

We study Finsler PL spaces, that is simplicial complexes glued out of simplices cut off from some normed spaces. We are interested in the class of Finsler PL spaces featuring local uniqueness of geodesics (for complexes made of Euclidean…

Let $G$ be a higher-rank semisimple Lie group over a nonarchimedean local field, for example $G={\rm PGL}(n,Q_P)$. To any lattice $L$ in $G$ there is an associated simplicial complex $B_L$, given by the quotient by $L$ of the Bruhat-Tits…