6. Geometric and Spectral Consequences of Curvature Bounds on Tessellations

This chapter focuses on geometric and spectral consequences of curvature bounds. Several of the results presented here have analogues in Riemannian geometry but in some cases one can go even beyond the Riemannian results and there also striking differences. The geometric setting of this chapter are tessellations and the curvature notion arises as a combinatorial quantity which can be interpreted as an angular defect and goes back to Descartes. First, we study the geometric consequences of curvature bounds. Here, a discrete Gauss–Bonnet theorem provides a starting point from which various directions shall be explored. These directions include analogues of a theorem of Myers, a Hadamard–Cartan theorem, volume growth bounds, strong isoperimetric inequalities and Gromov hyperbolicity. Secondly, we investigate spectral properties of the Laplacian which are often consequences of the geometric properties established before. For example we present analogues to a theorem of McKean about the spectral gap, a theorem by Donnelly-Li about discrete spectrum, we discuss the phenomena of compactly supported eigenfunctions and briefly elaborate on stability of the ℓ ² spectrum for the Laplacian on ℓ ^p .