Geometric Group Theory: Groups of Polynomial Growth

Math 835, Fall 2008

Geometric group theory is the study of finitely generated groups using geometry. Usually this means studying the geometry of the Cayley graph of a group with respect to some finite set of generators. This is a graph whose vertices are the group elements and whose edges correspond to multiplication on the right by a generator.

The growth function of a group is a function which describes how many group elements appear in a ball of any given radius in the Cayley graph. Although this function depends on a choice of generating set, its large-scale properties do not. It therefore makes sense to say that a group has (for instance) linear, quadratic, or exponential growth function.

A theorem of Gromov states:

A finitely generated group G has polynomial growth if and only if G has a finite index subgroup which is nilpotent.

This is one of the first examples of a theorem with an essentially geometric hypothesis and a strong algebraic conclusion.

Our main goal this semester will be to give a complete proof of Gromov's theorem. Along the way, we will learn about a number of tools (including nonstandard analysis) which have had a large impact on geometric group theory and related fields. Topics to be touched on include: Ultrafilters and nonstandard analysis, Lie groups and their subgroups, the structure of locally compact groups, coarse and quasi-isometric geometry, infinite nilpotent and solvable groups, amenability and Property (T).

Prerequisites for the material we will cover are extremely mild. You should know some basic group theory and point set topology.

References

• Misha Kapovich's notes contain an account of much of what we'll talk about in this class, and much else besides.
• A general reference for combinatorial group theory: Lyndon and Schupp, Combinatorial Group Theory, MR1812024.
• Information on length spaces can be found in: Burago, Burago, and Ivanov, A Course in Metric Geometry, MR1835418, or in Bridson and Haefliger, Metric Spaces of Non-Positive Curvature, MR1744486. (The book by Bridson and Haefliger is a treasure-trove, and if geometric group theory is interesting to you at all, you'll eventually want a copy.)
• For a nice survey of quasi-isometric rigidity, see Cornelia Druţu, Quasi-isometric rigidity of Groups, arXiv:math/0612256v2.
• The approach we are taking is basically that in: Van den Dries and Wilkie, "Gromov's theorem on groups of polynomial growth and elementary logic", Journal of Algebra 89 (1984) 349--374.
• A more leisurely introduction to nonstandard analysis can be found in: Goldblatt, Lectures on the hyperreals MR1643950 .
• More to come...