Dimer spaces and gliding systems

TL;DR

This paper introduces the dimer space of a graph as a nonpositively curved cubed complex, analyzes its fundamental group, and explores connections with Artin and braid groups, extending classical graph theory concepts.

Contribution

It develops the concept of dimer spaces as cubed complexes, proves their nonpositive curvature, and relates their fundamental groups to known algebraic structures.

Findings

Dimer space is a nonpositively curved cubed complex.

Fundamental group of the dimer space has a specific presentation.

Connections established with right-angled Artin and braid groups.

Abstract

Dimer coverings (or perfect matchings) of a finite graph are classical objects of graph theory appearing in the study of exactly solvable models of statistical mechanics. We introduce more general dimer labelings which form a topological space called the dimer space of the graph. This space turns out to be a cubed complex whose vertices are the dimer coverings. We show that the dimer space is nonpositively curved in the sense of Gromov, so that its universal covering is a CAT(0)-space. We study the fundamental group of the dimer space and, in particular, obtain a presentation of this group by generators and relations. We discuss connections with right-angled Artin groups and braid groups of graphs. Our approach uses so-called gliding systems in groups designed to produce nonpositively curved cubed complexes.