  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.
