2012/10/13 by Dave Witte Morris, Morris, Dave Witte
Computer Science · Mathematics · #22E40 #57S25 #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #math.GR #msc:22E40 #msc:57S25 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1210.3671
44 pages. Preliminary version - comments welcome
arxiv created 2012/10/13 · openalex publication_date 2012/10/13 · arxiv updated 2012/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.