Some arithmetic groups that are not left orderable

It is known that finite-index subgroups of the arithmetic group SL(3,Z) are not left orderable. (In other words, they have no interesting actions on the real line.) This naturally led to the conjecture that most other arithmetic groups (of higher real rank) also are not left orderable. The problem remains open, but joint work with Lucy Lifschitz verifies the conjecture for many examples, including every finite-index subgroup of SL(2,Z[sqrt(3)]) or SL(2,Z[1/3]). 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.