THURSTON'S BOUNDARY FOR TEICHMULLER SPACES OF INFINITE SURFACES: THE LENGTH SPECTRUM

Let X-0 be an infinite area geodesically complete hyperbolic surface which can be decomposed into geodesic pairs of pants. We introduce Thurston's boundary to the Teichmuller space T(X-0) of the surface X-0 using the length spectrum analogous to Thurston's construction for finite surfaces. Thurston's boundary using the length spectrum is a "closure" of projective bounded measured laminations PMLbdd(X-0), and it coincides with PMLbdd(X-0) when X-0 can be decomposed into a countable union of geodesic pairs of pants whose boundary geodesics {alpha(n)}(n is an element of N) have lengths pinched between two positive constants. When a subsequence of the lengths of the boundary curves of the geodesic pairs of pants {alpha(n)}n converges to zero, Thurston's boundary using the length spectrum is strictly larger than PMLbdd(X-0).