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.

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About metric2011

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