L3

We classify 2-dimensional polygonal complexes that are simply connected, platonic (in the sense that they admit a flag-transitive group of symmetries) and simple (in the sense that each vertex link is a complete graph).  These are a natural generalization of the 2-skeleta of simple polytopes.

Our classification is complete except for some existence questions for complexes made from squares and pentagons.

(Joint with Tadeusz Januszkiewicz, Raciel Valle and Roger Vogeler.)

A polygonal complex $X$ is Platonic if its automorphism group $G$ acts transitively on the flags (vertex, edge, face) in $X$. Compact examples include the boundaries of Platonic solids.  Noncompact examples $X$ with nonpositive curvature (in an appropriate sense) and three polygons meeting at each edge were classified by \'Swi\c{a}tkowski, who also determined when the group $G=Aut(X)$, equipped with the compact-open topology, is nondiscrete.  For example, there is a unique $X$ with the link of each vertex the Petersen graph, and in this case $G$ is nondiscrete.  A Fuchsian building is a two-dimensional also determined when the group $G=Aut(X)$, equipped with the compact-open topology, is nondiscrete.  For example, there is a unique $X$ with the link of each vertex the Petersen graph, and in this case $G$ is nondiscrete.  A Fuchsian building is a two-dimensional hyperbolic building.  We study lattices in automorphism groups of Platonic complexes and Fuchsian buildings.  Using similar methods for both cases, we construct uniform and nonuniform lattices in $G=Aut(X)$.  We also show that for some $X$ the set of covolumes of lattices in $G$ is nondiscrete, and that $G$ admits lattices which are not finitely generated.  In fact our results apply to the larger class of Davis complexes, which includes examples in dimension > 2.