Some arithmetic groups that do not act on the circle

TL;DR

This paper investigates why certain arithmetic groups, including SL(3,Z), cannot act faithfully on the circle, using tools from group theory such as orderability, amenability, and bounded cohomology.

Contribution

It introduces new methods to prove that specific arithmetic groups do not admit faithful actions on the circle or the real line, expanding understanding of group actions.

Findings

SL(3,Z) cannot act on the circle

Groups of the form SL(2,Z[a]) do not act on the real line

Actions on the circle have finite orbits

Abstract

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 as ordered groups, amenability, bounded generation, and bounded cohomology. Lecture 1 provides an introduction to the subject, and uses the theory of left-orderable groups to prove that SL(3,Z) does not act on the circle. Lecture 2 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 to amenable groups and bounded cohomology, respectively. They also explain how these 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 City Mathematics Institute's Graduate Summer School in July 2012.