The purpose of the course is to familiarize listeners with key concepts and methods of the geometric group theory. In particular, we present ideas of ​​the theory of groups acting on trees (Bass-Serre theory), quasi-isometric maps of spaces, hyperbolic metric spaces and hyperbolic groups in the sense of Gromov. Participants do not need special prerequisites, knowledge of the university course of algebra, as well as of the initial concepts of topology is enough. Students of any year and post-graduate students are invited.

One lecture per week is assumed. The duration of the lecture is 2 hours.

The program

Topic 1. Free groups.
Definition of a free group. Diamond lemma. Normal form of elements. Universal property of free groups

Topic 2. Group presentations.
Deductions using defining relations of a group. Algebraic interpretation of group presentations using quotients of free groups. Tietze transformations. Van Kampen diagrams. Van Kampen lemma.

Topic 3. Introduction to algorithmic problems.
Word and conjugacy problems. The conjugacy problem in free groups. The isomorphism problem. An example of a class of groups with a solvable isomorphism problem: finitely generated Abelian groups. The membership problem in a subgroup. Unsolvability of most algorithmic problems for groups (without proof).

Topic 4. Graphs in geometric group theory.
Graphs as combinatorial 1-complexes. Trees. The fundamental group of a graph. Coverings of graphs. Group actions: basic concepts. Cayley and Schreier graphs. Schreier theorem on subgroups of a free group. The membership problem for a free group.

Topic 5. Asymptotic characteristics of groups.
The word metric on a group. The growth function. Invariance of the growth function with respect to the choice of a generating set. The Dehn function. Invariance of the Dehn function with respect to the choice of the presentation. Examples of upper bounds for the Dehn function.

Topic 6. Free constructions.
Free products of groups. The normal form of elements of a free product. Universal property of a free product. Free products with amalgamation. Modified version of the diamond lemma. The normal form of elements of a free product with amalgamation. Universal property of a free product with amalgamation. HNN extensions of groups. The normal form of elements of an HNN extension. Britton's lemma.

Topic 7. Introduction to the theory of groups acting on trees.
Graphs of groups. Construction of the fundamental group of a graph of groups. Construction of a group action tree for free constructions. Construction of a group action tree in the general case.

Topic 8. Introduction to rough geometry.
Quasi-isometric embeddings. Quasi-isometries. Quasi-isometricity criteria. Milnor-Schwarz theorem. Quasi-isometric invariants of groups.

Topic 9. Hyperbolic metric spaces.
Geodesic spaces. Metric trees. Equivalent definitions of a hyperbolic metric space.

Topic 10. Hyperbolic groups.
The Gromov theorem on the equivalence of the hyperbolicity of a groups and the linearity of the Dehn function. Examples of hyperbolic groups: discrete subgroups of isometries of a hyperbolic space; groups with a small cancellation condition.

Lecturer
Lysenok Igor Geront'evich

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center