Related papers: Dimer spaces and gliding systems
The minimum dominating set problem asks for a dominating set with minimum size. First, we determine some vertices contained in the minimum dominating set of a graph. By applying a particular scheme, we ensure that the resulting graph is…
We study Witt groups of smooth curves and surfaces over algebraically closed fields of characteristic not two. In both dimensions, we determine both the classical Witt group and Balmer's shifted Witt groups. In the case of curves, the…
There are theories of coverings of $C^*$-algebras which can be included into a following list: coverings of commutative $C^*$-algebras, coverings of $C^*$-algebras of groupoids and foliations, coverings of noncommutative tori, the double…
We examine distortion of finitely generated normal subgroups. We show a connection between subgroup distortion and group divergence. We suggest a method computing the distortion of normal subgroups by decomposing the whole group into…
We study here some aspects of the topology of the space of smooth, stable, genus 0 curves in a Riemannian manifold $X$, i.e. the Kontsevich stable curves, which are not necessarily holomorphic. We use the Hofer-Wysocki-Zehnder polyfold…
In recent years, discrete spaces such as graphs attract much attention as models for physical spacetime or as models for testing the spirit of non-commutative geometry. In this work, we construct the differential algebras for graphs by…
We announce results on the structure of CAT(0) groups, CAT(0) lattices and of the underlying spaces. Our statements rely notably on a general study of the full isometry groups of proper CAT(0) spaces. Classical statements about Hadamard…
This paper provides a comprehensive study of the dimer model on infinite minimal graphs with Fock's elliptic weights [arXiv:1503.00289]. Specific instances of such models were studied in [arXiv:052711, arXiv:1612.09082, arXiv1801.00207]; we…
Croke and Kleiner constructed two homeomorphic locally CAT(0) complexes whose universal covers have visual boundaries that are not homeomorphic. We construct two homeomorphic locally CAT(0) complexes so that the visual boundary of one…
Directions of graphs were originally introduced in the study of a cops-and-robbers kind of game, while the study of end spaces has been used to generalize classical graph-theoretical results to infinite graphs, such as Halin's…
In this project we explore the geometry of general metric spaces, where we do not necessarily have the tools of differential geometry on our side. Some metric spaces $(X,d)$ allow us to define geodesics, permitting us to compare geodesic…
This article investigates the properties of order-divisor graphs associated with finite groups. An order-divisor graph of a finite group is an undirected graph in which the set of vertices includes all elements of the group, and two…
An arithmetical structure on a finite, connected graph without loops is an assignment of positive integers to the vertices that satisfies certain conditions. Associated to each of these is a finite abelian group known as its critical group.…
We introduce a correspondence between dimer models (and hence superconformal quivers) and the quantum Teichmuller space of the Riemann surfaces associated to them by mirror symmetry. Via the untwisting map, every brane tiling gives rise to…
Snake graphs are a class of planar graphs that are important in the theory of cluster algebras. Indeed, the Laurent expansions of the cluster variables in cluster algebras from surfaces are given as weight generating functions for 1-dimer…
Theoretical results from discrete geometry suggest that normed spaces can abstractly embed finite metric spaces with surprisingly low theoretical bounds on distortion in low dimensions. In this paper, inspired by this theoretical insight,…
Isoradial graphs are a natural generalization of regular graphs which give, for many models of statistical mechanics, the right framework for studying models at criticality. In this survey paper, we first explain how isoradial graphs…
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 partition functions for the dimer model on families of finite graphs converging to infinite self-similar graphs and forming approximation sequences to certain well-known fractals. The graphs that we consider are provided by actions…
Based on our previous graph covering method, we introduce weighted graph covering models and flexible graph covering models that are almost equivalent to the well-known Bratteli--Vershik models. These models play important roles in showing…