## Notes of Andrew Sale’s Cambridge lecture 12-01-2017

When RAAG have vast outer automorphism groups

Joint with Vincent Guirardel.

1. RAAGs in Bridson’s universe of groups

From ${{\mathbb Z}}$, 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 ${G}$ is large if it maps onto ${F_2}$. ${Gl(n,{\mathbb Z})}$ is large only if ${n=2}$. ${Out(F_n)}$ is large for ${n=2,3}$. Grünewald-Lubotzky have constructed a virtual rational representation virtually onto ${Sl((n-1)h,{\mathbb Z})}$ for all ${n\geq 2}$ and ${h\in{\mathbb Z}}$. This gives largeness for ${n=3}$.

Definition 1 Say ${G}$ has all finite groups involved (AFGI) if every finite group ${F}$ is a quotient of some finite index subgroup of ${G}$.

Then ${Out(F_n)}$ has AFGI for all ${n}$, by Grünewald-Lubotzky.

2. Generators of ${Out(A)}$

Goes back to Lawrence 1995. Let ${A}$ be a RAAG. The following elements generate ${Out(A)}$:

1. Partial conjugation. Fix a vertex ${v}$, conjugate by ${v}$ outside the star of ${v}$, identity on the star.
2. Transvection. Fix vertices ${u,v}$. Map ${v}$ to ${uv}$ and do not change other generators, get maps ${L_u^v}$ (resp. ${R_u^v}$). Some conditon is necesary. Define a partial preorder on vertices: ${v\leq u}$ iff ${link(v)\subseteq star(u)}$. Then ${L_u^v}$ and ${R_u^v}$ are automorphisms.
3. Inversion ${v\mapsto v^{-1}}$.
4. Graph symmetries.

Say ${u}$ is equivalent to ${v}$ if ${u\leq v}$ and ${v\leq u}$. If so, they have the same links. Let ${[u]}$ denote an equivalence class. Two cases: ${A_{[u]}}$ is a free group or a free abelian group.

Theorem 2 (Day 2011) ${Out(A)}$ contains ${F_2}$ iff either

1. ${\Gamma}$ contains an equivalence class of size at least 2.
2. There is a separating intersection of links (SIL).

Otherwise, ${Out(A)}$ is virtually nilpotent.

A SIL is a triple ${(x,t|z)}$ of vertices such that ${d(x,y)\geq 2}$ and every path from ${x}$ to ${y}$ must path through ${link(x)\cap link(y)}$. Then partial conjugations by ${x}$ and ${y}$ generate a free group.

3. Result

Theorem 3 (Guirardel-Sale) ${OUT(A)}$ has AFGI iff

1. ${\Gamma}$ contains a free equivalence class of size at least 2.
2. ${\Gamma}$ contains an equivalence class of size 2.
3. ${\Gamma}$ has a SIL.

Otherwise, there is a short exact sequence ${1\rightarrow B\rightarrow Out(A)\rightarrow \prod Sl(n_i,{\mathbb Z})\rightarrow 1}$, where ${N}$ is finitely generated and nilpotent. Here, ${n_i\geq 3}$ are the sizes of equivalence classes.

The first lemma is that ${Out(A)}$ surjects onto ${Out(A_{[u]})}$ for each equivalence class ${[u]}$. This handles cases 1 and 2.

For case 3, given a SIL ${(x_1,x_2|x_3)}$, let ${S}$ be the union of the equivalence classes of ${x_1,x_2,x_3}$. Then ${Out(A)}$ surjects onto ${Out(A_S)}$. Game with partial conjugations and transvections.

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