## 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.

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).