** When RAAG have vast outer automorphism groups **

Joint with Vincent Guirardel.

**1. RAAGs in Bridson’s universe of groups **

From , two branches divide: amenable branch, free and hyperbolic branch. In between sit right-angles Artin groups. Beyond is the realm if the lion.

What can one say about the outer automorphism groups ?

Say a group is *large* if it maps onto . is large only if . is large for . Grünewald-Lubotzky have constructed a virtual rational representation virtually onto for all and . This gives largeness for .

**Definition 1** * Say has all finite groups involved (AFGI) if every finite group is a quotient of some finite index subgroup of . *

Then has AFGI for all , by Grünewald-Lubotzky.

**2. Generators of **

Goes back to Lawrence 1995. Let be a RAAG. The following elements generate :

- Partial conjugation. Fix a vertex , conjugate by outside the star of , identity on the star.
- Transvection. Fix vertices . Map to and do not change other generators, get maps (resp. ). Some conditon is necesary. Define a partial preorder on vertices: iff . Then and are automorphisms.
- Inversion .
- Graph symmetries.

Say is equivalent to if and . If so, they have the same links. Let denote an equivalence class. Two cases: is a free group or a free abelian group.

**Theorem 2 (Day 2011)** * contains iff either *

*
*
- contains an equivalence class of size at least 2.
- There is a separating intersection of links (SIL).

* Otherwise, is virtually nilpotent. *

A SIL is a triple of vertices such that and every path from to must path through . Then partial conjugations by and generate a free group.

**3. Result **

**Theorem 3 (Guirardel-Sale)** * has AFGI iff *

*
*
- contains a free equivalence class of size at least 2.
- contains an equivalence class of size 2.
- has a SIL.

* Otherwise, there is a short exact sequence , where is finitely generated and nilpotent. Here, are the sizes of equivalence classes. *

The first lemma is that surjects onto for each equivalence class . This handles cases 1 and 2.

For case 3, given a SIL , let be the union of the equivalence classes of . Then surjects onto . Game with partial conjugations and transvections.

We need to generalize Grünewald-Lubotzky to treat outer automorphism groups of free products of free abelian groups.

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## About metric2011

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