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The group SL(3,Z) cannot act (faithfully) on the circle (by homeomorphisms).We will see that many other arithmetic groups also cannot act on the circle.The discussion will involve several important topics in group theory, such asordered groups, amenability, bounded generation, and bounded cohomology.  Lecture 1 provides an introduction to the subject, and uses the theory ofleft-orderable groups to prove that SL(3,Z) does not act on the circle. Lecture2 discusses bounded generation, and proves that groups of the form SL(2,Z[a])do not act on the real line. Lectures 3 and 4 are brief introductions toamenable groups and bounded cohomology, respectively. They also explain howthese ideas can be used to prove that actions on the circle have finite orbits.An appendix provides hints or references for all of the exercises.  These notes are slightly expanded from talks given at the Park CityMathematics Institute's Graduate Summer School in July 2012.

Some arithmetic groups that do not act on the circle