Analysis and Geometry on Groups

The
following classical problems in group theory will be discussed:

• Siegel problem: finite generation of lattices.
• Margulis construction: expander graphs.
• Atiyah invariants: L2 Betti numbers.
• Gromov problem: non-uniform exponential growth.
• Burnside groups: infinite, finitely generated torsion groups.
• Day class: exotic amenable groups.

Along presenting solutions to these problems we will introduce the following
subjects:

• Amenability and property (T).
• Random groups and graphs.
• Groups generated by automata.
• L2 invariants of groups and manifolds.
• Random walks on groups and graphs.