FALL 2006 MATH 595 Section IGG
MWF, 10am Altgeld Hall, rm 441
WWW:
http://www.math.uiuc.edu/~kapovich/595-06/595-06.html
Instructor:
Ilya Kapovich
Telephone:
265-0633
e-mail: kapovich@math.uiuc.edu. (Preferred
method
of reaching me!)
Office location: Altgeld Hall, room 365
Office hours (preliminary): Monday
1:30pm-3:00pm,
Thursday, 10am-11:30am AND by appointment
Text:
There is no official required textbook
for this course.
Recommended books 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)
Other recommended sources (this list will be augmented over time):
- Topics in Combinatorial Group Theory, by G. Baumslag, Lectures in Mathematics (ETH Zurich), Birkhauser, 1993
- Metric Spaces of
Non-positive Curvature, by M. Bridson and A. Haefliger,
Springer, 1999
- Groups Acting on Graphs, by W. Dicks and M. Dunwoody, Cambridge studies in advanced mathematics, vol. 17, Cambridge University Press, 1989
- J. Stallings, Topology of
finite graphs, Invent. Math. 71 (1983), no. 3, pp. 551-565
- I. Kapovich and A. Myasnikov, Stallings foldings and subgroups of free
groups, Journal of Algebra 248
(2002), pp. 608-668
Prerequisites: Math 500 "Abstract Algebra I". Some
familiarity with the basics of algebraic topology, such as
fundamental groups and covering spaces, is desirable.
Brief course description.
and deeper roots in which groups were studied primarily from
combinatorial and topological perspectives. In the mid 1980's,
spurred by ideas of Michael Gromov, group theorists began to pay
more attention to the interplay between algebraic properties of
finitely generated groups and geometric properties of spaces on
which the groups act. This attention shed a great deal of light
on the earlier combinatorial and topological investigations into
group theory, and stimulated other innovative ideas which have
been developing at a rapid pace. Many adjacent fields have been
substantially affected, such as low-dimensional topology,
hyperbolic geometry and Riemannian geometry, Kleinian groups,
lattices in Lie groups, geometric analysis, logic, complexity
theory, etc.
This will be an introductory course whose purpose is to give some
idea about what geometric group theory is and to provide the
background knowledge for those students who may be interested in
working in this area later.
At the end of the course
I will expect each student to make a
20-25 minute presentation
on a paper (I will provide a number of choices and you can also choose
a paper not from my list that will have to be approved by me) that
concerns some recent developments in Geometric
Group Theory. Additionally, you will have to explain in detail some of
the proofs from your paper of choice to me individually.
Approximate
Syllabus.
- Free groups and their subgroups via Stallings subgroup graphs.
- Groups given by generators and relators. Cayley graphs and the word metric. Van Kampen diagrams and van Kampen Theorem. Small cancellation conditions and Dehn's algorithm. CAT(0)-cubical complexes.
- Amalgamated free products and HNN-extensions. Some application (e.g. Adian-Rabin Theorem, embeddability into two-generator groups, counter-examples to Hopficity and co-Hopficity, 2- and 3-dimensional topology examples). Groups acting on simplicial trees and Bass-Serre theory.
- Quasi-isometries, geometric properties and
invariants. Ends, growth, isoperimetric functions, amenability,
solvability of
the word problem and asymptotic cones as examples of geometric invariants. - Word-hyperbolic groups: definitions, examples, and
basic
properties. - One or two advanced topics, such as: the
Novikov-Boone Theorem, Burnside groups, Grigorchuk groups of
intermediate
growth, automatic groups, relative hyperbolicity, actions on R-trees, Thompson's group, the Culler-Vogtmann outer
space, genericity and random groups, etc.
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