TOPICS IN GEOMETRIC GROUP THEORY
FALL 2005  MATH 595 Section TGT
MWF, 10am    Altgeld Hall, rm 445

WWW:     http://www.math.uiuc.edu/~kapovich/595-05/595-05.html

Telephone: 265-0633
e-mail: kapovich@math.uiuc.edu. (Preferred method of reaching me!)
Office location: Altgeld Hall, room 365
Office hours (preliminary):   Tuesday 4:00-5:30pm, Thursday, 10am-11:30am  AND by appointment
Text:   There is no official required textbook for this course.
             Recommended sources are:

•  Topics in Geometric Group Theory, by P. de la Harpe, University of
  Chicago Press, 2000
•  Cominatorial Group Theory, by R. Lyndon and P. Schupp,
  Springer-Verlag, 2001; ("Classics in Mathematics series'', reprint of the 1977 edition)
•  Lecture notes on Geometric Group Theory [pdf file], by Michael Kapovich (be aware that there are lots of misprints in this file); a postscript version is available here.
  

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Brief course description.

The first part of the course will be devoted to proving Gromov's
theorem about characterizing finitely generated groups of polynomial
growth, which is one of the most beautiful and important results in
Geometric Group theory in the last 30 years.

We will then cover a number of topics related to growth of groups and
to volume entropy, such as amenability of finitely generated groups
(including its various geometric, analytic and probabilistic
characterizations) and random walks on graphs and groups.  We will
discuss some of the modern developments in this area related to
uniform amenability, uniformly exponential growth, growth tightness,
topological amenability, Kazhdan's property T, etc.  Time permitting we will consider in
more detail some important examples, such as Thompson's groups and
Grigorchuk's groups of intermediate growth.
 
There are no formal prerequisites for this course but the participants
are expected to be somewhat familiar with such notion as free groups,
groups given by generators and relators, smooth manifolds and Lie
groups.
Nevertheless, I will try to teach this course in a way that does not assume much advanced knowledge and is suitable as an introductory course to the subject.

At the end of the course I will expect each student to make a presentation on some paper (I will provide a number of choices) that is reasonably short, self-contained and yet concerns some recent developments in Geometric Group Theory.  You can also  choose  a paper not from my list, but I'll have to approve it first.