Foliation-Preserving Maps Between Solvmanifolds
Copyright policy )Foliation-Preserving Maps Between Solvmanifolds
For i = 1,2, let Gamma_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 Gamma_igX_i provide a foliation F_i of the homogeneous space Gamma_i\G_i. Let f be a continuous map from Gamma_1\G_1 to Gamma_2\G_2 whose restriction to each leaf of F_1 is a covering map onto a leaf of F_2. If we assume that F_1 has a dense leaf, and make certain technical technical assumptions on the lattices Gamma_1 and Gamma_2, then we show that f must be a composition of maps of two basic types: a homeomorphism of Gamma_1\M_1 that takes each leaf of F_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 G_1 and G_2 are neither solvable nor semisimple.
- Williams College United States
lattice subgroup, Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\), Nilpotent and solvable Lie groups, Geometric Topology (math.GT), affine map, Homogeneous spaces, solvable Lie group, Mathematics - Geometric Topology, foliation, homogeneous space, FOS: Mathematics, Foliations in differential topology; geometric theory
lattice subgroup, Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\), Nilpotent and solvable Lie groups, Geometric Topology (math.GT), affine map, Homogeneous spaces, solvable Lie group, Mathematics - Geometric Topology, foliation, homogeneous space, FOS: Mathematics, Foliations in differential topology; geometric theory
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