4/5/2005

Dave Witte Morris
University of Lethbridge

Some arithmetic groups that cannot act on the line

It is known that finite-index subgroups of the arithmetic group SL(3,Z) have no interesting actions on the real line.  This naturally led to the conjecture that most other arithmetic groups (of higher real rank) also cannot act on the line.  This problem remains open, but joint work with Lucy Lifschitz verifies the conjecture for many examples.  This includes all finite-index subgroups of SL(2,Z[a]) or SL(2,Z[1/n]), where a is any real, irrational algebraic integer, and n > 1. The proofs are based on the fact,  proved by D.Carter, G.Keller, and E.Paige, that every element of these groups is a product of a bounded number of elementary matrices.