Non-left orderability of lattices in higher rank Lie groups

A countable group is said to be  left-orderable if it preserves a total order invariant by left multiplication or equivalently  if it embeds in the group of homeomorphisms of the line. I’ll explain the basics of left-orderability and what is known about left-orderability of lattices in Lie groups. 

 

Our main result is that an irreducible lattice in a real semi-simple Lie group G of higher rank is left-orderable if and only if G is a product of two Lie groups and one factor is the universal covering of SL_2(R). In particular, we show that every lattice in SL_n(R) (if n > 2) is not left-orderable, solving conjectures of Witte-Morris and Ghys. The tools used in the proof include 1) the study of random walks by homeomorphisms in the line,  2) the construction of a compactification of a group action in the line and 3) the study of the stiffness of some stationary measures. (Joint work with Bertrand Deroin).