GEOMETRIC GROUPS AND ANALYSIS

MATH 582G - WINTER 2010

INSTRUCTOR INFORMATION:

Instructor: Steffen Rohde
Office: Padelford C-337, Phone: (206) 543 6171
E-mail: rohde@math.washington.edu
Office hours: TBA

Instructor: Boris Solomyak
Office: Padelford C-328, Phone: (206) 685-1307
E-mail: solomyak@math.washington.edu
Office hours: Mondays 4-5, Tuesdays 11:30-12:30 in C-328, or by appointment

 

COURSE DESCRIPTION: This course will be an introduction, from an analyst's perspective, into a fascinating area which links Groups, Geometry, Analysis, Dynamics, and Probability. The groups we are interested in are discrete and infinite, with finitely many generators and relations. From this data we get the Cayley graph of a group, which is studied as a metric space. Some topics will be worked out in detail, whereas others will be presented in overview (in particular, we will sometimes quote results from other areas without proof). There will be lots of examples.

The PREREQUISITES will be kept at a minimum. In particular, we will not assume any knowledge of Group Theory beyond the very basics, and all concepts from algebra, geometry, topology, analysis or probability (such as presentations, fundamental group, hyperbolic geometry, harmonic functions, random walk) will be defined and reviewed.

USEFUL LINKS:

• B. Bowditch "Course in Geometric Group Theory"
• M. Kapovich "Lectures on Geometric Group Theory"
• B. Kleiner "A new proof of Gromov's theorem on groups of polynomial growth"
• Y. Shalom and T. Tao "A finitary version of Gromov's polynomial growth theorem"
• T. Tao "On Kleiner's proof of Gromov's Theorem" (from the blog)

 

RESERVE LIST (though we will mostly follow the online articles):

• Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, ed. by Bedford et al
• Topics in geometric group theory, by Pierre de La Harpe
• Sur les groups hyperboliques d'apres Mikhael Gromov, by Ghys, de la Harpe
• Groups, graphs, and trees, by John Meier

TENTATIVE PLAN

PART I (1st week taught by B. Solomyak, then by S. Rohde)

• Finitely-generated groups, Cayley graphs as metric spaces, examples.
• Free groups, "Ping-pong" lemma (how to check when a subgroup is free).
• Quasi-isometries.
• Elements of hyperbolic geometry.
• Surface groups.
• Gromov hyperbolic groups.
• Boundaries, Cannon's conjecture (presented in outline).

PART II (taught mostly by B. Solomyak; some classes taught by S. Rohde + guest lectures by Asaf Nachmias)

• Growth of groups (polynomial, exponential, intermediate) -- overview.
• Gromov's Theorem on groups of polynomial growth: discussion, scheme of the proof.
• Liouville property (every bounded harmonic function is constant) for groups of subexponential growth.
• Proof of Gromov's Theorem (following B. Kleiner).
• Amenability (including the discussion of the Banach-Tarski paradox).
• Random walks on groups.