__Beginning Lecture Course__

**Lecturers: **Tara Brendle (Cornell University)

Indira Chatterji (Cornell University)

**Teaching Assistants: **Pallavi Dani (University of Chicago)

Talia Fernos (University of Illinois at Chicago)

The course will begin with an introduction to the fundamentals of metric spaces and groups. We will see how groups are metric spaces and explore specific examples, with an eye on so-called word hyperbolic groups. Another focus will be braid groups, which naturally carry a rich geometrical structure, as well as related groups such as Artin groups, Coxeter groups, and mapping class groups.

Specific topics might be distributed as follows (subject to change according to the background of the audience):

1. Metric spaces

2. Cayley graphs

3. Group presentations

4. Fundamental groups

5. Braid groups I

6. Introduction to surfaces

7. Braid groups II

8. Hyperbolic groups

__Beginning Lecture Course Suggested Background Reading__

Basic group theory, as covered in Artin's "Algebra" or Fraleigh's "A First Course in Abstract Algebra," for example.

__Advanced Lecture Course__

**Lecturers: **1st week - Ruth Charney (Brandeis University)

2nd week - Karen Vogtmann (Cornell University)

**Teaching Assistants: **Angela Barnhill (The Ohio State University) for Ruth Charney

Emina Alibegovic (University of Michigan) for Karen Vogtmann

Beginning with the work of Dehn, geometric group theory has studied the structure of groups via their actions on metric spaces. Classical problems in geometric group theory include algorithmic problems, such as the word and conjugacy problem, and questions about the structure of subgroups. In the first part of the course, we will discuss these and other problems in the context of CAT(0) spaces, and particularly CAT(0) cube complexes. In the second part of the course, we will explore groups acting on trees and spaces of trees.

- Classical problems in geometric group theory
- Geometric actions of groups and quasi-isometry
- CAT(0) spaces
- Cube complexes and application
- Groups acting on trees
- Groups acting on spaces of trees
- Compactifying spaces of trees

__Advanced Lecture Course Reading List__

**Week 1:**

Bridson, M., Non-positive curvature in group theory. *Groups St. Andrews 1997 in Bath, I, *124--175, London Math. Soc. Lecture Note Ser., 260, *Cambridge Univ. Press, Cambridge, *1999. (an expository article)

Bridson, M., Geodesics and curvature in metric simplicial complexes. *Group theory from a geometrical viewpoint (Trieste, 1990), *373--463, *World Sci. Publishing, River Edge, NJ, *1991. (sections 2-3 give a good introduction to CAT(0) geometry)

Cannon, J., Geometric group theory. *Handbook of geometric topology, *261--305, *North-Holland, Amsterdam, *2002. (a broad survey of topics in geometric group theory)

Charney, R., Metric geometry: connections with combinatorics. *Formal power series and algebraic combinatorics (New Brunswick, NJ, 1994), *55--69, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 24, *Amer. Math. Soc., Providence, RI, *1996. (an expository article)

Niblo, G., and Reeves, L., The geometry of cube complexes and the complexity of their fundamental groups. *Topology ***37 **(1998), no. 3, 621--633. (a good introduction to cube complexes)

**Week 2:**

Bestvina, M., $\\Bbb R$-trees in topology, geometry, and group theory. *Handbook of geometric topology, *55--91, *North-Holland, Amsterdam, *2002. (a survey)

Bestvina, M. and Feighn, M., Notes on Sela\'s work: Limit groups and Makhanin-Razborov diagrams, available at http://www.math.utah.edu/~bestvina/research.html

Scott, P. and Wall, T., Topological methods in group theory. *Homological group theory (Proc. Sympos., Durham, 1977), *pp. 137--203, London Math. Soc. Lecture Note Ser., 36, *Cambridge Univ. Press, Cambridge-New York, *1979. (an expository article)

Serre, J.-P., Trees. Springer Monographs in Mathematics. *Springer-Verlag, Berlin, *2003. (first half of the book gives the basics of groups acting on trees)

Vogtmann, K., Automorphisms of free groups and outer space. Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000). *Geom. Dedicata *94 (2002), 1--31. (a survey)