Related papers: Dimer spaces and gliding systems

In this work we describe a class of subsets of the Euclidean plane which, with the induced length metric, are locally $CAT(0)$ spaces and we show that the gluing of two such subsets along a piece of their boundary is again a locally…

The dimer model on a graph embedded in the torus can be interpreted as a collection of random self-avoiding loops. In this paper, we consider the uniform toroidal honeycomb dimer model. We prove that when the mesh of the graph tends to zero…

Let M be a graph manifold. We show that \pi_1M is the fundamental group of a compact nonpositively curved cube complex if and only if M is chargeless. We also prove that in that case \pi_1M is virtually compact special.

A dimer model is a quiver with faces embedded into a disk. A consistent dimer model gives rise to a strand diagram, and hence to a positroid. The Gorenstein-projective module category over the completed boundary algebra of a dimer model was…

We express the finite 3D Dimer partition function as a linear combination of determinants of oriented adjacency matrices, and the finite 3D Ising partition sum as a linear combination of products over aperiodic closed walks. The methodology…

We provide a systematic description of the automorphism groups of specially cocompact CAT(0) cube complexes. We show that these groups are topologically finitely generated, present a method to explicitly obtain generating sets, and prove a…

We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the CAT(k) inequality. We prove a general CAT(k)…

k-graphs are higher-rank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of Cuntz-Krieger type. Here we develop the theory of covering spaces for k-graphs, obtaining a…

Covering spaces are a fundamental tool in algebraic topology because of the close relationship they bear with the fundamental groups of spaces. Indeed, they are in correspondence with the subgroups of the fundamental group: this is known as…

The kth finite subset space of a topological space X is the space exp_k X of non-empty finite subsets of X of size at most k, topologised as a quotient of X^k. The construction is a homotopy functor and may be regarded as a union of…

We introduce two new tools for studying CAT(0) spaces: \emph{curtains}, versions of cubical hyperplanes; and the \emph{curtain model}, a counterpart of the curve graph. These tools shed new light on CAT(0) spaces, allowing us to prove a…

We present a general homotopical analysis of structured diagram spaces and discuss the relation to symmetric spectra. The main motivating examples are the I-spaces, which are diagrams indexed by finite sets and injections, and J-spaces,…

Orbifold and logarithmic structures provide independent routes to the virtual enumeration of curves with tangency orders for a simple normal crossings pair $(X|D)$. The theories do not coincide and their relationship has remained…

We study the dimer model for a planar bipartite graph N embedded in a disk, with boundary vertices on the boundary of the disk. Counting dimer configurations with specified boundary conditions gives a point in the totally nonnegative…

In this paper we define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an $n$-dimensional cube to a fixed metric space. We…

We study quasiisometric embeddings between finite-dimensional CAT(0) cube complexes. More specifically, we introduce geometric branching conditions under which flats in the domain, not necessarily of top rank, are mapped within finite…

A matching $M$ in a graph $\Gamma$ is positive if $\Gamma$ has a vertex-labeling such that $M$ coincides with the set of edges with positive weights. A positive matching decomposition (pmd) of $\Gamma$ is an edge-partition $M_1,\ldots,M_p$…

Universal coverings of the orthogonal groups and their extensions are studied in terms of Clifford-Lipschitz groups. An algebraic description of basic discrete symmetries (space inversion $P$, time reversal $T$, charge conjugation $C$ and…

We consider new cosmological solutions which generalize the cosmological patch of the Anti-de Sitter (AdS) space-time, allowing for fluids with equations of state such that $w\neq -1$. We use them to derive the associated full manifolds. We…

We develop the boundary theory of rough CAT(0) spaces, a class of spaces that contains both Gromov hyperbolic and CAT(0) spaces. The resulting theory generalizes the common features of the Gromov boundary of a Gromov hyperbolic space and…