Twist Surfaces

In this paper, we consider the problem: what kinds of Riemann surfaces S will have rst eigenvalue 1 (S) large? Bye large, we mean that there is some positive lower bound for S independent of the genus. There are a number of constructions of surfaces with 1 large, the main example being the quotients of the hyperbolic plane by the congruence subgroups of PSL(2; Z). Most of the known examples rely on fairly heavy number-theoretic or representation-theoretic machinery. The construction of SX] replaces much of the representation theory with geometric arguments, but still only applies to surfaces with some special types of symmetries. On the other hand, it is our belief that \large rst eigenvalue" should be a much more common phenomenon than this, although it seems to be diicult to pinpoint good geometric criteria which will allow for this. The technique we will employ involves thinking of a Riemann surface in terms of a graph. Then a surface will have large rst eigenvalue if and only if the corresponding graph has large rst eigenvalue. The novelty of our approach is that we will associate a graph to a surface in such a way that many surfaces will share the same graph. The surfaces which share the same graph, which we will call twist surfaces, will in general have very little in common with one another, except for the fact that they will have a comon lower bound for the rst eigenvalue. They will not even have the same genus! In this way, we may build from arithmetic examples large collections of very non-arithmetic examples which have their rst eigenvalue bounded from below. Our thinking on this question was inspired by the theorem of Joel Fried-man F], who showed that, for k even, a \typical" k-regular nite graph is close to being Ramanujan. One would hope, arguing by analogy, that a \typ-ical" Riemann surface has rst eigenvalue close to 1=4. While the present results are far from that goal, we hope to emphasize the role that nite graph theory can play in constructing fairly exotic surfaces whose spectral theory is nonetheless under good control. An important part of our argument revolves around the use of the Ahlfors-Schwarz Lemma. We believe that this could be a powerful tool in the relationship between graph theory and geometry, and hope the reader nds this discussion of independent interest. The results of this paper …