Syllabus
Basics of combinatorial group theory:
1. Free group, subgroups of a free group (the method of Nielsen and Reidemeister), the relationship between the index and the rank of subgroup of a group of a finite index, subgroups of finite rank as free factors in a subgroup of a finite index. Conjugation and cyclically reduced words.
2. Tietze transformations.
3. HNN extensions, defining relations, Britton's lemma and the normal form theorem, applications of HNN extensions.
4. Free products with an amalgamated subgroup, defining relations, the normal form theorem.
5. Geometrical methods, the fundamental group of a two-dimensional complex, application for a geometrical proof that a subgroup of a free group is free, Kurosh's theorem, Grushko -- von Neumann's theorem.
6. Cayley complexes According to an interest, some of the following topics will be tought.
1. Higman's embedding theorem.
2. Small cancellation theory.
3. Braid group, the word problem, factors, connections to authomorphisms of free groups.
4. Groups acting on trees.
5. Hyperbolic groups.
6. Tessellations and Fuchsian complexes.
7. Solvability of the word problem for groups with one defining relation.
8. Bipolar structures.
Annotation
Subgroups of free groups (Nielsen's and Reidemaister's method), Tietze transformations, HNN extensions, free products with an amalgamated subgroup, geometrical methods, Cayley complexes. Other selected topics in elementary combinatorical group theory.