Sectional curvature of polygonal complexes with planar substructures
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 2dimensional Euclidean and hyperbolic buildings. We focus on the case of nonpositive and negative combinatorial curvature. As geometric results we obtain a HadamardCartan 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 DonnellyLi 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2014
 arXiv:
 arXiv:1407.4024
 Bibcode:
 2014arXiv1407.4024K
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Group Theory;
 Mathematics  Spectral Theory
 EPrint:
 32 pages, 3 figures