First lecture: January 15, 2002, 15.15; room 3733.
Schedule: The course consisted of 16 lectures
during weeks 3-7 and 9-11
(January 15 - February 15 and February 25 - March 15, 2002).
There were two lectures per week:
Tuesdays, 15.15-17.00, room 3733
Thursdays, 10.15-12.00, room 3721
Exam: The exam consists in solving problems from the
following
UPDATED (02.03.14, 16.30!) problem list
Deadline: 02.05.01
if you want it checked before the summer break
Topics discussed in the course:
Chapter I. FREE PRODUCTS OF GROUPS
1. Free products of groups.
2. Definition of the fundamental group of a topological space. Its
topological invariance.
Chapter II. FREE GROUPS AND PRESENTATIONS OF GROUPS
3. Free groups. Universal property of free products and of free groups.
4. Presentations of groups. Von Dyck's Theorem. Tietze
transformations.
M. Dehn's algorithmic problems.
5. Particular cases of Seifert - van Kampen Theorem. The fundamental
group of a bouquet of circles is free.
Chapter III. FREE GROUPS, TREES, AND GRAPHS
6. Graphs and trees. The fundamental group of a connected graph is free.
7. Cayley graphs. Actions of a group on a graph.
8. A group which acts freely on a tree is free. Nielsen-Schreier
Theorem: a subgroup of a free group is free.
9. Subgroups and coverings.
Howson's property for free groups
(following J. Stalling's notes
on Geometric Group Theory.)
10. Subgroups of free groups. Schreier graphs. Schreier's formula.
Chapter IV. FREE PRODUCTS WITH AMALGAMATION AND HNN-EXTENSIONS
11. Free products with amalgamation. Topological motivation. Surface
groups.
12. Higman-Neumann-Neumann Embedding Theorem. HNN-extensions.
Higman's example of a finitely generated but not finitely presentable
group.
13. Britton's Lemma. Finitely presented amalgamated products.
Uncountably many groups with two generators.
14. Residually finite groups. Hopfian and non-hopfian
groups. Amalgamated products and trees.
Chapter V. GROUPS AND GEOMETRY
15. Poincaré's Theorem. Hyperbolic tesselations.
16. A short introduction into Coxeter groups. Length functions, word
metrics, and growth of groups.
Recommended Literature:
W. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory
R. Lyndon, P. Schupp, Combinatorial Group Theory, Chapters I-IV
G. de Rham, Lectures on Introduction to Algebraic Topology
W. Massey, Algebraic Topology: an Introduction, Chapters 2-6
J. Rotman, An Introduction to the Theory of Groups, Chapters
1-2, 11-12
D. L. Johnson, Presentations of Groups
G. Baumslag, Topics in Combinatorial Group Theory, Chapters
I, III, VI
J. Stillwell, Classical Topology and Combinatorial Group
Theory, Chapters 0.5, 2, 3, 4, 5, 7
D. Cohen, Combinatorial Group Theory: a topological approach,
Chapters 1-2, 5, 9
J.-P. Serre, Trees, Chapters I.1-I.4
P. de la Harpe, Topics in Geometric Group Theory, Chapters II-V
B. Chandler, W. Magnus, The history of combinatorial group
theory. A case study in the history of ideas.
W. Magnus, Noneucledian tesselations and their groups
J. Humphreys, Reflection Groups and Coxeter Groups