Sectional curvature of polygonal complexes with planar substructures

Keller, Matthias; Peyerimhoff, Norbert; Pogorzelski, Felix

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.

Citation

Keller, M., Peyerimhoff, N., & Pogorzelski, F. (2017). Sectional curvature of polygonal complexes with planar substructures. Advances in Mathematics, 307, 1070-1107. https://doi.org/10.1016/j.aim.2016.10.027

• Journal Article Type: Article
• Acceptance Date: Oct 24, 2016
• Online Publication Date: Dec 6, 2016
• Publication Date: Feb 5, 2017
• Deposit Date: Oct 24, 2016
• Publicly Available Date: Oct 25, 2016
• Journal: Advances in Mathematics
• Print ISSN: 0001-8708
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 307
• Pages: 1070-1107
• DOI: https://doi.org/10.1016/j.aim.2016.10.027