Related papers: Cubulating spaces with walls

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 study topological median algebra structures on Euclidean spaces and, more generally, ER homology manifolds. We show that all such median structures have a local CAT(0) cubulation structure. We also show that topological median algebra…

We give an effective characterisation of the walls in the variation of geometric invariant theory problem associated to a quiver and a dimension vector.

In this work we introduce a new combinatorial notion of boundary $\Re C$ of an $\omega$-dimensional cubing $C$. $\Re C$ is defined to be the set of almost-equality classes of ultrafilters on the standard system of halfspaces of $C$, endowed…

The wall-and-chamber structure is a geometric invariant that can be associated to any algebra. In this notes we give the definition of this object and we explain its relationship with torsion classes and $\tau$-tilting theory.

We consider relationships between cubic algebras and implication algebras. We first exhibit a functorial construction of a cubic algebra from an implication algebra. Then we consider an collapse of a cubic algebra to an implication algebra…

A tubular group is a group that acts on a tree with $\mathbb{Z}^2$ vertex stabilizers and $\mathbb{Z}$ edge stabilizers. This paper develops further a criterion of Wise and determines when a tubular group acts freely on a finite dimensional…

We formulate a relation between quantum-mechanical coherent states and complex-differentiable structures on the classical phase space ${\cal C}$ of a finite number of degrees of freedom. Locally-defined coherent states parametrised by the…

Petyt-Spriano-Zalloum recently developed the notion of a \textit{curtain model}, which is a hyperbolic space associated to any CAT(0) space. It plays a similar role for CAT(0) spaces that curve graphs do for mapping class groups of…

Heinrich Tietze has shown that for a closed connected subset of euclidean space being convex is a local property. We generalize this to CAT(0)-spaces and locally compact CAT(\kappa) spaces. As an application we give a construction of…

In this article, we associate to isometries of CAT(0) cube complexes specific subspaces, referred to as \emph{median sets}, which play a similar role as minimising sets of semisimple isometries in CAT(0) spaces. Various applications are…

We show that a certain triangulation of CAT(0) triangle-pentagon complexes is $7$-located and locally $5$-large. Hereby we give examples of $7$-located, locally $5$-large groups.

We prove that, under certain mild conditions, every cocompact CAT(0) space is almost geodesically complete.

For each $n$ we construct examples of finitely presented $C'(1/6)$ small cancellation groups that do not act properly on any $n$-dimensional CAT(0) cube complex.

In this paper, we study the geometry of cone-offs of CAT(0) cube complexes over a family of combinatorially convex subcomplexes, with an emphasis on their Gromov-hyperbolicity. A first application gives a direct cubical proof of the…

A class of Cantor-type spaces and related geometric structures are discussed.

Ballmann's Rank Rigidity Conjecture predicts that a CAT(0) space of higher rank with a geometric group action is rigid -- isometric to a Riemannian symmetric space, a Euclidean building, or splits as a direct product. We confirm this…

The notion of a coherent space is a nonlinear version of the notion of a complex Euclidean space: The vector space axioms are dropped while the notion of inner product is kept. Coherent spaces provide a setting for the study of geometry in…

We study the collapsibility of finite simplicial complexes of dimension 3 endowed with a CAT(0) metric. Our main result states that, under an additional hypothesis, finite simplicial 3-complexes endowed with a CAT(0) metric collapse to a…

We discuss thick domain walls interpolating between spaces with naked singularities and give arguments based on the $AdS$/CFT correspondence why such singularities may be physically meaningful. Our examples include thick domain walls with…