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Abstract:
In the early 80's Gromov initiated a program to study
finitely generated groups up to quasi-isometry. This program was
motivated by rigidity properties of lattices in Lie groups. A
lattice $\Gamma$ in a group $G$ is a discrete subgroup where the
quotient $G/\Gamma$ has finite volume. Gromov's own major theorem
in this direction is a rigidity result for lattices in nilpotent
Lie groups.
In the 1990's, a series of dramatic results led to the completion
of the Gromov program for lattices in semisimple Lie groups. The
next natural class of examples to consider are lattices in
solvable Lie groups, and even results for the simplest examples
were elusive for a considerable time.
In around 2005, Eskin, Whyte and I introduced a technique of
coarse differentiation that led to the first quasi-isometric
rigidity results for lattices in solvable Lie groups. I will
describe something about these results and techniques and will also
talk about what is involved in extending the results to more general
solvable Lie groups. Some of this is work of Irine Peng and some is joint
work with Eskin and Peng.
I will also describe some interesting results concerning groups
quasi-isometric to homogeneous graphs that follow from the same
methods.
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