Dimer spaces and gliding systems

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.