Sectional curvature of polygonal complexes with planar substructures

Matthias Keller,Norbert Peyerimhoff,Felix Pogorzelski

doi:10.1016/j.aim.2016.10.027

Abstract

In this paper we introduce a class of polygonal complexes for which we consider a notion of sectional combinatorial curvature. These complexes can be viewed as generalizations of 2-dimensional Euclidean and hyperbolic buildings. We focus on the case of non-positive and negative combinatorial curvature. As geometric results we obtain a Hadamard–Cartan type theorem, thinness of bigons, Gromov hyperbolicity and estimates for the Cheeger constant. We employ the latter to get spectral estimates, show discreteness of the spectrum in the sense of a Donnelly–Li type theorem and present corresponding eigenvalue asymptotics. Moreover, we prove a unique continuation theorem for eigenfunctions and the solvability of the Dirichlet problem at infinity.

Highlights

Since recent years there is an increasing interest in studying curvature notions on discrete spaces
We introduce a notion of sectional curvature for more general non-planar polygonal complexes
We identify a class of polygonal complexes that is well suited for our purposes

Summary

Introduction

Since recent years there is an increasing interest in studying curvature notions on discrete spaces. We introduce sectional curvatures on the faces and on the corners of an apartment (see Definition 2.8), and they are invariants measuring the local geometry of the polygonal complex with planar substructures. We present a combinatorial Cartan-Hadamard theorem for non-positively curved polygonal complexes with planar substructures (see Theorem 3.1) and we conclude Gromov hyperbolicity and positivity of the Cheeger isoperimetric constant for negatively curved polygonal complexes with planar substructures with certain bounds on the vertex and face degree (see Theorems 3.6 and 3.8) These results are based on negativity or non-positivity of the sectional corner curvature. 2-dimensional Euclidean and hyperbolic buildings provide large classes of examples of polygonal complexes with planar substructures While all these spaces have non-positive sectional face curvature, their corner curvature is not always necessarily non-positively curved. In the appendix we discuss how Wise’s definition of sectional curvature, which in some sense an even more flexible notion, is related to our notion of curvature

Basic definitions

Geometry

Spectral theory

Examples