GROUPS ACTING ON TREES
FALL 2002  MATH 415 Section B1
MWF, 9am    Altgeld Hall, rm 241

WWW:     http://www.math.uiuc.edu/~kapovich/415-02/415-02.html

Telephone: 265-0633
e-mail: kapovich@math.uiuc.edu. (Preferred method of reaching me!)
Office location: Illini Hall, room 328
Office hours (preliminary):   Tuesday, Thursday, 10am-11:30am  AND by appointment
 

TEXTBOOK:  There is no single assigned textbook for this course.
The primary sources we shall use are:
 
 

• G.Baumslag, "Topics in Combinatorial Group Theory", Lectures in Mathematics, Birkhauser, 1993
• L.Bartholdi, R.Grigorchuk and Z.Sunik, "Branch Groups",  in preparation (copies of manuscript will be supplied by the instructor)

Other basic references include:
 

• J.-P. Serre, "Trees", Springer-Verlag, 1980
• R.Lyndon and P.Schupp, "Combinatorial Group Theory", Springer, 1977 and a reprint in Classic in Mathematics series, 2001
• D.Cohen, "Combinatorial Group Theory: A Topological Approach", London Mathematical Society Student Texts 14, 1989
• H.Bass, "Covering theory for graphs of groups", Journal of Pure and Applied Algebra, 89 (1993), no. 1-2, pp. 3-47
• M.Dunwoody, "Cutting up graphs", Combinatorica 2 (1982), no. 1,  pp. 15-23
• E. Rips and Z.Sela,  "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition", Annals  of Mathematics (2) 146 (1997), no. 1,  pp. 53-109

---

Brief course description.

In this course we will cover two main topics: Bass-Serre theory of groups acting on simplicial trees and the theory of branch groups (groups acting on rooted trees).

We will start with a review of such basic notions as free groups, groups given by generators and relators, free products with amalgamations and HNN-extensions. We will then proceed to cover in detail the foundational results of Bass-Serre theory. After that we will prove in detail one major theorem: Stallings' theorem about ends of groups. We will also discuss in some generalities some recent important applications of the Bass-Serre theory, such as the JSJ-decomposition of finitely presented groups.

Regarding the branch groups, we will first go over their definitions and basic properties. We will then discuss some important examples of applying the brach groups machinery: Grigorchuk groups of intermediate growth, automata groups, Burnside groups etc.

I will try to keep the exposition algebraic and will use topological considerations only as a source of intuition.

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 demonstrates an applications of the theory of groups acting on trees.