grantor:
University of TorontoIn the late '80s, A. Lubotzky, R. Phillips and P. Samak [22] used Deligne's proof of Ramanujan's conjecture and the Jacquet-Langlands correspondence for cuspidal representations of 'GL'(2) to construct a class of Ramanujan graphs which are the best known expander graphs. Their graphs are Cayley graphs of the group 'PSL' &parl0;2,Z/qZ&parr0; or 'PGL' &parl0;2,Z/qZ&parr0; . Unfortunately, this strategy cannot be applied for groups in general because, in the general case, there is no equivalent of the Jacquet-Langlands correspondence. However, J. Rogawski has completely classified the representations of the unitary group in three variables in [25]. In this thesis we consider a form of 'U'(3) which at a place p is isomorphic to 'GL'3 &parl0;Qp&parr0; . We study a finite disconnected union of finite quotients of the building attached to the group 'GL'3 &parl0;Qp&parr0; . We view this object as a hypergraph and, using the classification of automorphic representations of the group 'U'(3) and Deligne's Theorem, we give an estimation of the spectrum of the adjacency matrix of its underlying graph. We show that the underlying graph is an expander graph with good expansion coefficient.Ph.D
