Amandine Escalier - Measure and orbit equivalence of graph products

Abstract

Graph products were introduced by Green in 1990 and generalise both right-angled Artin groups and right-angled Coxeter groups. Like the latter two, graph products act nicely on CAT(0) cube complexes, making them particularly appealing objects for geometric group theorists.

Although CAT(0) cube complexes and geometric group theory are absolutely fantastic, here, we will study graph products from the equally fantastic point of view of measured dynamics. All necessary prerequisites will be recalled, the words “measured groupoids” might be pronounced (but only to give a name to colourful pictures), and parallels with geometric group theory will be made.

This (the results presented, not the abstract) is a joint work with Camille Horbez.

Talk

Goals: Review into measured dynamics using Artin and Coxeter groups

Measure equivalence is the analogue of quasi-isometry. For fintie Out QI and ME of RAAGs actually match.

Motivation

Definition

We can show that $G$ and $H$ are isomorphic iff. they act freely, the 2 actions commute and they are both transitive.

Definition

Two groups are quasi-isometric iff there is a locally compact space, they act on s.t.

• they are properly discontinuous
• the 2 actions commute
• they both admit a compact fundamental domain

Definition

$G,H$ are Measure equivalent groups if there exists a space they act on a space st.

• freely, measure preserving
• the 2 actions commute
• each admit a fundamental domain of finite measure

Definition

$G,H$ are Orbit equivalent groups if they are ME with a common fundamental domain

Example

$F_2$ and $F_3$ are measure equivalent but not orbit equivalent.

Theorem

All infinite countable Amenable Group are orbit equivalent to $\mathbb{Z}$. e.g. ${\mathbb{Z}}^d$, Baumslag-Solitar Gruppen, $\mathbb{Z}/2\mathbb{Z}$

Theorem (Kida 06)

If $G$ is measure equivalent like a Mapping class group, then $G$ is almost a MCG itself

Theorem (Guirardel-Horbez, 21)

A similar thing but for the Outer automorphisms of free group.

• I want to learn a topic, that connects all of the mentioned groups. What should I read?

Definition

Let $\Gamma$ be a Graph and $G_v$ be a family of groups, one for each vertex. The Graph product is the free products, with commutators between elements of two groups iff. the two groups are connected by an edge.

This generalised Freies Produkt and Direct product.

The case of RAAGs and RACGs

Theorem

Let $W_n$ be the RACG over the $n$-gon. For all $n\ge 5$ all $W_n$ are measure equivalent.

Theorem

Two RAAGs with finite Äußerer Automorphismusgroup are isomorphic iff. they are ME iff. their defining graphs are the same.

We want to find criterions for which the out if the RAAGs is finite. It is finite it is has no Transvections and Partial conjugations.

Theorem (E.-Horbez 24)

If we take two graph with out transvections or partial conjugations and contably infinite groups at every vertex. Then they are ME iff. they are OE iff. there is a graph isomorphism such that all associated groups on the vertices are orbit equivalent.

Note that finite automorphism is actually needed.

3. Tools

3.1. Extension graphs

Definition

We define the Extension graph by choosing as edges some conjugated elements and adding some edges by some rule.

We showed some very important result but I didn’t pay attention anymore.