  Let \hat{S} be the algebraic universal cover of a closed surface of genus >1,
T(\hat{S}) its Teichmuller space, M(\hat{S}) the group of mapping classes
stabilizing a fixed leaf l. The L^1 Ehrenpreis conjecture asserts that
M(\hat{S}) on T(\hat{S}) with dense orbits with respect to the L^1 topology
(the topology induced by the L^1 norm on quadratic differentials). We give a
proof of this weaker form of the Ehrenpreis conjecture.
