2024/2025
Gromov Hyperbolic Groups
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
1, 2 module
Open to:
students of all HSE University campuses
Instructors:
Alexey Golota
Language:
English
ECTS credits:
6
Course Syllabus
Abstract
Historically, the study of infinite groups was primarily motivated by problems from geometry. On the other hand, geometric methods are widely used to explore the structure of groups. For instance, for a group with a finite set of generators one can define its Cayley graph and a natural left-invariant metric on it. In 1980s M. Gromov laid the foundations of a theory of <<hyperbolic>> groups, that is, the groups with Cayley graphs of <<negative curvature>> (in an appropriate sense). The class of hyperbolic groups is quite large, for example, it includes lattices in Lie groups of rank 1, fundamental groups of negatively-curved manifolds, free groups etc. The theory of hyperbolic groups is a rich theory with numerous applications. The goal of the course is to provide an introduction to this theory and, more generally, to study methods of geometric group theory via examples.
Course Contents
- Metric geometry: metric spaces, isometries, geodesics
- Basic notions of combinatorial group theory: Cayley graphs, group actions on trees
- CAT-inequalities, CAT(0)-spaces, Cartan – Hadamard theorem
- Equivalent definitions of hyperbolic groups, examples
- Basic properties of hyperbolic groups
- Quasi-isometries of metric spaces, quasi-geodesics
- The boundary of a hyperbolic space, actions of isometries on the boundary
- More general actions on hyperbolic spaces: relative hyperbolicity, hyperbolically embedded subgroups
Interim Assessment
- 2024/2025 2nd moduleThe final grade for the course consists of the grades for problem sheets and the grade for a take-home exam.
