“Tukia’s theorem and the large scale geometry certain solvable groups”

Tukia’s theorem states that any uniform group of quasiconformal maps
of the sphere that acts cocompactly on the space of distinct triples
of points can be conjugated into the conformal group. This theorem is
used as step in the proof of quasi-isometric rigidity for lattices in
real hyperbolic spaces.

Recently Eskin-Fisher-Whyte have described quasi-isometries of certain
solvable Lie groups. Lattices in these Lie groups form classes of
polycyclic groups. I will formulate and sketch the proof of a
Tukia-like theorem which can be used to prove quasi-isometric rigidity
for these lattices.