Notes of Bertrand Deroin’s lecture

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 {G} is a lattice in a simple Lie group of real rank {\geq 2}, then {G} is not left-orderable.

Theorem 1 (Witte) True for finite index subgroups of {SL(n,{\mathbb Z})}, {n\geq 3}.

Conjecture (Linnell): Let {G} be a finitely generated left-orderable group. Then either {G} contains {F_2}, or {G} surjects onto {{\mathbb Z}}.

Theorem 2 (Witte) Let {G} be a finitely generated amenable left-orderable group. Then {G} surjects onto {{\mathbb Z}}.

1.1. Actions on the real line

Proposition 3 Countable left-orderable groups coincide with subgroups of Homeo{^+({\mathbb R})}.

Idea of proof: enumerate elements of {G} and map them to {{\mathbb R}} in an order preserving manner. Left translations act on a subset of {{\mathbb R}} and extend by continuity to homeos.

2. Actions on the circle

Theorem 4 (Ghys) If {G} is a lattice in a simple Lie group of real rank {\geq 2}, then every actions of {G} on the circle by homeomorphisms has a finite orbit.

Theorem 5 (Ghys) If {G\subset Homeo^+(S^1)} has no finite orbit, then either {G} contains {F_2}, or {G} surjects onto {{\mathbb Z}}.

2.1. Quasi-periodic actions on the real line

Can one upgrade left-orderable (action on {S^1} with a fixed point) to action on {S^1} without finite orbits ? One can do a step in this direction.

Definition 6 {\rho:G\rightarrow Homeo^+({\mathbb R})} is quasi-periodic if its conjugates by translations form a relatively compact subset in {Hom(G,Homeo^+({\mathbb R}))}.

A source of such actions is the following situation. Let {X} be a compact space with a free action of {{\mathbb R}} and a {G}-action mapping {{\mathbb R}}-orbits to {{\mathbb R}}-orbits. Conversely, every quasi-periodic action on {{\mathbb R}} arises in this way.

3. Link with random walks

Let {\mu} be a symmetric probability distribution on {G} whose finite support generates {G}. Assume {G\subset Homeo^+({\mathbb R})}. This defines a Markov process on {{\mathbb R}}, where {p(x,y)=\mu\{g\in G\,;\,gx=y\}}.

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: {\forall x\in{\mathbb R}}, the average displacement {\mathop{\mathbb E}_{x,\mu} (gx)=x}. Moreover, this action is quasi-periodic.

3.1. Proof

Uses recurrence. There exists a compact interval {K\subset {\mathbb R}} such that for all {x\in{\mathbb R}}, almost surely, the trajectory {g_n\ldots g_1 x} hits {K} infinitely often.

From recurrence, we get existence of a stationary Radon measure on {{\mathbb R}} (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/
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