## Some arithmetic groups that are not left orderable

- Date: 03/19/2007

Dave Morris (University of Lethbridge)

University of British Columbia

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.

Algebraic Geometry Seminar 2007