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.