Random walks on left-orderable groups
I view random walks as a tool to study left-orderable groups.
1. Left-orderable groups
This means a group with a left-invariant order.
Torsion in an obstruction. Not easy to find firther obstructions, and indeed, many classes of groups are left-orderable.
- torsion free nilpotent groups (Malcev),
- free groups (Magnus),
- surface groups,
- virtually, every 3-manifold group (Agol).
Conjecture: If is a lattice in a simple Lie group of real rank , then is not left-orderable.
Theorem 1 (Witte) True for finite index subgroups of , .
Conjecture (Linnell): Let be a finitely generated left-orderable group. Then either contains , or surjects onto .
Theorem 2 (Witte) Let be a finitely generated amenable left-orderable group. Then surjects onto .
1.1. Actions on the real line
Proposition 3 Countable left-orderable groups coincide with subgroups of Homeo.
Idea of proof: enumerate elements of and map them to in an order preserving manner. Left translations act on a subset of and extend by continuity to homeos.
2. Actions on the circle
Theorem 4 (Ghys) If is a lattice in a simple Lie group of real rank , then every actions of on the circle by homeomorphisms has a finite orbit.
Theorem 5 (Ghys) If has no finite orbit, then either contains , or surjects onto .
2.1. Quasi-periodic actions on the real line
Can one upgrade left-orderable (action on with a fixed point) to action on without finite orbits ? One can do a step in this direction.
Definition 6 is quasi-periodic if its conjugates by translations form a relatively compact subset in .
A source of such actions is the following situation. Let be a compact space with a free action of and a -action mapping -orbits to -orbits. Conversely, every quasi-periodic action on arises in this way.
3. Link with random walks
Let be a symmetric probability distribution on whose finite support generates . Assume . This defines a Markov process on , where .
Theorem 7 (Deroin-Kleptsyn-Navas-Parwani) Assume that orbits are dense. Then there exists a conjugation (unique up postcomposition by affine transformations) such that the Deriennic property holds: , the average displacement . Moreover, this action is quasi-periodic.
Uses recurrence. There exists a compact interval such that for all , almost surely, the trajectory hits infinitely often.
From recurrence, we get existence of a stationary Radon measure on (Ornstein-Weiss).