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.
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.