Foliation-Preserving Maps Between Solvmanifolds

For i=1,2, let _i be a lattice in a simply connected, solvable Lie group G_i, and let X_i be a connected Lie subgroup of G_i. The double cosets _igX_i provide a foliation _i of the homogeneous space _iG_i. Let f be a continuous map from ₁G₁ to ₂G₂ whose restriction to each leaf of ₁ is a covering map onto a leaf of ₂. If we assume that ₁ has a dense leaf, and make certain technical assumptions on the lattices ₁ and ₂, then we show that f must be a composition of maps of two basic types: a homeomorphism of ₁G₁ that takes each leaf of ₁ to itself, and a map that results from twisting an affine map by a homomorphism into a compact group. We also prove a similar result for many cases where G₁ and G₂ are neither solvable nor semisimple.