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Let 7V be a complete hyperbolic w-manifold and C(N) its convex core. The manifold N is said to be geometrically finite if the metric neighborhood Q (7V) of radius one about C(7V) is of finite volume (see Bowditch [2] for equivalent definitions). The Laplace-Beltrami operator Δ of N acts on the space of C°°-functions with compact support and admits a unique extension to an unbounded self-adjoint operator on L(N). Let λ0(Ν) denote the bottom of the L-spectrum of — Δ. (Notice that λ0 (N) ^ 0.) When N is geometrically finite, λ0 (N) = 0 if and only if 7V has finite volume. Here we are interested in infinite volume geometrically finite hyperbolic manifolds and our main result is a lower bound on λ0(Ν) in terms of the volume of Q (7V), provided n ̂ 3.

A lower bound on λ 0 for geometrically finite hyperbolic n-manifolds.