Lie Groups, Representations and Discrete Mathematics

Paley graphs are well-known combinatorial objects which have many interesting properties. Many of these properties come from their symmetry under the automorphisms x-->ax+b of the affine line over a finite field F with q elements (q=4m+1). We construct new simplicial complexes attached to certain groups, concentrating on G=PGL(2,F_q) for any finite field F. For every divisor d of q-1 we construct a "small" 2-dimensional complex with G-action. In a special case, the star of each vertex is a Paley graph. When d dereases from q to 1 the fundamental groups of the complexes give free, surface, property T, building, and finally simply connected complexes (this uses the Weil bounds for the number of points on curves over a finite field)