We prove that an infinite Riemann surface X is parabolic (X∈OG) if
and only if the union of the horizontal trajectories of any integrable
holomorphic quadratic differential that are cross-cuts is of zero measure. Then
we establish the density of the Jenkins-Strebel differentials in the space of
all integrable quadratic differentials when X∈OG and extend Kerckhoff's
formula for the Teichm\"uller metric in this case. Our methods depend on
extending to infinite surfaces the Hubbard-Masur theorem describing which
measured foliations can be realized by horizontal trajectories of integrable
holomorphic quadratic differentials.Comment: 41 pages, 9 figure