Abstract

Using the Levi decomposition theorem, Lie groups are usually studied in two separate classes: semisimple and solvable. Both these classes further divide into two subclasses with very different behaviour: semisimple groups split into the rank 1 and higher rank cases; while solvable groups divide into those of polynomial growth and those of exponential growth.

Amongst connected unimodular Lie groups, let us say that G is “small” if it shares a cocompact subgroup with some direct product of a rank one simple Lie group and a solvable Lie group with polynomial growth. Otherwise, we say G is “large”. We present three strong dichotomies which distinguish “small” and “large” groups; which are respectively algebraic, coarse geometric, and local analytic in nature. As an application we will show that Baumslag-Solitar groups admit a similar “small”/”large” dichotomy. This is part of a joint project with John Mackay and Romain Tessera.