*Ma 191e* - Geometry of Infinite Groups; Winter 2003

### Instructor: Danny Calegari

### TuTh 3-4:30 153 Sloan

### Grading policy:

Those taking the class for a grade will be required to take notes.

### Description of course:

In this class we will study discrete groups as geometric objects.
We are particularly interested in the geometry "in the large" of
the Cayley graphs of finitely presented groups, and the ways in which
certain algebraic properties of these groups are manifested in geometric ways.
We will start with an introduction to certain important classes of
groups - hyperbolic groups, lattices, groups acting on circles,
groups acting on trees, mapping class groups - and some of their basic
(geometric) properties. If there is time, we hope to discuss the material
in Gromov's recent preprint "Random walk in random groups," available
from Gromov's home page at IHES.

### Topics to be covered (if time allows):

- Quasi-isometric and lipschitz geometry
- Important examples of groups
- Hyperbolic groups and coarse negative curvature
- Isoperimetric functions and algorithmic theory of groups
- Amenability, property T and a-T-menability
- Random groups

Notes for the course will gradually be posted in installments.
The first installment is posted here.

### References:

- D. Epstein et. al.
*Word processing in groups* Jones and Bartlett (1992)
- M. Gromov
*Hyperbolic groups* in: Essays in group theory, pp. 75-263, MSRI publications vol. 8 (1987)
- M. Gromov
*Asymptotic invariants of infinite groups* LMS lecture note series vol. 182 (1993)
- M. Gromov
*A random walk in random groups* preprint, available at IHES
- S. Katok
*Fuchsian groups* Chicago Lectures in Mathematics, University of Chicago Press (1992)
- A. Lubotzky
*Discrete groups, expanding graphs and invariant measures* Birkhauser Progress in Mathematics 123 (1994)
- K. Ohshika
*Discrete groups* AMS translations of Mathematical Monographs 207 (1998), translated from the Japanese by the author
- J-P. Serre
*Trees* Springer-Verlag (1980), translated from the French by John Stillwell
- R. Zimmer
*Ergodic theory and semisimple groups* Birkhauser Monographs in Mathematics 81 (1984)