** 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. *

** 3.1. Proof **

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).

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## About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
http://www.math.ens.fr/metric2011/