Zeta Functions for Hyperbolic Surfaces
نویسنده
چکیده
Let M = H/Γ a “convex co-compact” hyperbolic surface consisting of a compact core and, possibly, a finite number of infinite volume funnels. We do not allowM to have cusps. (More formally, Γ is convex co-compact if (convex.hull(LΓ))/Γ is compact, where LΓ is the limit set of Γ.) We exclude the possibility that Γ is elementary (i.e. virtually cyclic). The closed geodesics on M are is one-to-one correspondence with the non-trivial conjugacy classes in π1M ∼= Γ. Let CG denote the (countably infinite) set of closed geodesics on M and let PCG denote the set of prime closed geodesics on M (i.e. those which are not multiples of another closed geodesic. We write lγ for the length of γ ∈ CG. For each x ≥ 0, #{γ ∈ PCG (or CG) : lγ ≤ x} is finite. We shall consider the zeta function
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