Bucolic Complexes

TL;DR

This paper introduces bucolic complexes, a new class generalizing systolic and CAT(0) cubical complexes, exploring their properties and implications in geometric group theory and topology.

Contribution

It defines bucolic complexes, characterizes them via their skeleta, and proves their contractibility and nonpositive-curvature-like properties.

Findings

Bucolic complexes generalize known classes like systolic and CAT(0) complexes.

They are contractible and exhibit nonpositive-curvature-like properties.

Characterizations via 2-skeleta and 1-skeleta enable broader understanding of their structure.

Abstract

We introduce and investigate bucolic complexes, a common generalization of systolic complexes and of CAT(0) cubical complexes. They are defined as simply connected prism complexes satisfying some local combinatorial conditions. We study various approaches to bucolic complexes: from graph-theoretic and topological perspective, as well as from the point of view of geometric group theory. In particular, we characterize bucolic complexes by some properties of their 2-skeleta and 1-skeleta (that we call bucolic graphs), by which several known results are generalized. We also show that locally-finite bucolic complexes are contractible, and satisfy some nonpositive-curvature-like properties.