A geometric lower bound on the (2g-2)th eigenvalue of a hyperbolic surface of genus g.

The existence of hyperbolic surfaces with small eigenvalues (eigenvalues less than 1/4) was originally
proved by B. Randol. P. Buser used a different method to show the existence of such surfaces in any
genus. Buser's construction showed the existence of closed hyperbolic surfaces of genus g with 2g-3
small eigenvalues. It is proved recently by Jean-Pierre Otal and Eulalio Rosas that 2g-3 indeed is the
maximal number i.e. for a genus g closed hyperbolic surface the (2g-2)th eigenvalue is strictly greater
than 1/4. In this talk we shall prove an explicit lower bound for the (2g-2)th eigenvalue for a genus g
closed hyperbolic surface depending only on the systole (or injectivity radius) of the surface.