Zeta Functions of Group Based Graphs and Complexes

In this survey article, we focus on graphs and complexes arising from quotients of the Bruhat-Tits buildings associated to PGL(2)(F) and PGL(3)(F), respectively. As such, the combinatorial objects, like vertices, edges and chambers, are parametrized algebraically by cosets, and the combinatorial adjacency operators can be interpreted as operators supported on suitable double cosets acting on certain L(2)-spaces. The algebraic structure provides links to group theory and number theory. We show interesting connections between combinatorics and number theory, mainly through zeta functions.