Sectional curvature of polygonal complexes with planar substructures

2014/07/15 by Matthias Keller, Keller, Matthias, Norbert Peyerimhoff +3
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG) #Spectral Theory (math.SP) #math.GR #math.MG #math.SP

paper · pdf · doi:10.48550/arxiv.1407.4024

32 pages, 3 figures

arxiv created 2014/07/15 · arxiv updated 2014/07/16

Abstract

In this paper we introduce a class of polygonal complexes for which we can define 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.

Related

• Curvature, geometry and spectral properties of planar graphs
• A combinatorial negative curvature condition implying Gromov\n hyperbolicity
• Combinatorial negative curvature and triangulations of three-manifolds
• Geometric and spectral properties of locally tessellating planar graphs
• A combination theorem for combinatorially non-positively curved\n complexes of hyperbolic groups