Bucolic Complexes

Details Bucolic Complexes Bucolic Complexes

2012-02-06

arxiv.org/abs/1202.1149v1

Mathematics

Combinatorics

37 pages, 4 figures

Scientific paper

In this article, we introduce and investigate bucolic complexes, a common generalization of systolic complexes and of CAT(0) cubical complexes. This class of complexes is closed under Cartesian products and amalgamations over some convex subcomplexes. We study various approaches to bucolic complexes: from graph-theoretic and topological viewpoints, as well as from the point of view of geometric group theory. Bucolic complexes can be defined as locally-finite simply connected prism complexes satisfying some local combinatorial conditions. We show that bucolic complexes are contractible, and satisfy some nonpositive-curvature-like properties. In particular, we prove a version of the Cartan-Hadamard theorem, the fixed point theorem for finite group actions, and establish some results on groups acting geometrically on such complexes. We also characterize the 1-skeletons (which we call bucolic graphs) and the 2-skeletons of bucolic complexes. In particular, we prove that bucolic graphs are precisely retracts of Cartesian products of locally finite weakly bridged graphs (i.e., of 1-skeletons of weakly systolic complexes). We show that bucolic graphs are exactly the weakly modular graphs satisfying some local conditions formulated in terms of forbidden induced subgraphs and that finite bucolic graphs can be obtained by gated amalgamations of products of weakly bridged graphs.}

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