Elia Fioravanti - Automorphisms of virtually special groups

Abstract

Mapping class groups, $Out(F_n)$ (Outer automorphisms of free group) and $G{L}_n(Z)$ are three important groups in geometric group theory that arise as outer automorphism groups of other groups (respectively, of surface, free and free abelian groups). In general, it can be rather complicated to describe the structure of the group $Out(G)$ for more general groups $G$, particularly if only certain properties of $G$ are known. I will outline some of the work of Rips and Sela in the 90s, which completely describes $Out(G)$ for any Gromov-hyperbolic group $G$, based on a canonical $JSJ$-decomposition of $G$. Then I will talk about a new canonical $JSJ$ decomposition for the compact special groups of Haglund and Wise, and how this can be used to gain information on $Out(G)$ for any virtually Special group $G$.

Talk

Remark: $X$ aspherical Cell complex. Topologically, we might want to study the spacce of homotopy equivalences up to homotopy.

Motivations: What is the structure of $Out$? None in general!

Theorem (Bungin-Wise '85)

Any countable group arises as $Out(G)$ for some $G$.

• Is there merit in studying an Out(G)-hierarchy?
• The Out(G) can be any arbitrary group. But Out(G) of hyperbolic groups are very rigid. Where does the complexity come from. It comes from flats interacting in complicated patterns. (Maybe the Tree of flats is an example)

Example

• $G={\mathbb{Z}}^n$: $Out({\mathbb{Z}}^n)=G{L}_n(\mathbb{Z})$
• $G={\pi }_{1}S$: $Out(G)=MCG(S)$ (Mapping class group) (Dehn-Hilsonn-Baer theorem?)
• $G=F_n$: $Out(F_n)$ (Outer automorphisms of free group)
• $G=BS(2,4)$: $Out(G)$ is not finitely generated (Baumslag-Solitar Gruppen)
• $G=V$: $Out(G)$ has bad automorphisms (Thompson’s group V)

They are the main motivation of most of GGT.

Hyperbolic groups

While in general understanding seems to be impossible, hyperbolic groups are different. They are very nice (and have been studied by Rips, Sela, Bestvina, Feighn, Bouditch)

We don’t unterstand hyperbolic groups but funnily, we do understand the outer automorphism!

Theorem

Let $G$ be Hyperbolic group and Torsionsfreie Gruppe.

1. If $Out(G)$ is infinite, then $G$ splits as $A{\ast }_{C}B$ or $A{\ast }_C$ where $C\cong \{1\}$ or $\mathbb{Z}$.
2. If $G$ is “1-ended” then it has a canonical $JSJ$-decomposition (it has an $Out(G)$-invariant splitting as a graph of groups) s.t.
   • all edge groups are $\cong \mathbb{Z}$
   • Vertex groups are either rigid (cant split further) or quadratically hanging (can split further but dont, because you want canonical).

Example

Let $G$ be the Fundamental group of some surface with a boundary. At the boundary we add many new surfaces (branching allowed). We also attach a $3$-space to a closed essential curve. Then the Automorphism group is simply a graph of groups where every vertex is a surface and they are connected just like the branched manifold

Corollary

Let $G$ be hyperbolic + torsion-free + 1-ended group. Then there is a $Ou{t}^0\subseteq Out(G)$ f.i. subgroup s.t. there is a central extension $1\to \mathbb{Z}\to Ou{t}^0(G)\to MCG(S_1,\partial S_1)\times ...\times MCG(S_1,\partial S_1)\to 1$

• openquestion Characterise $Out(G)$ for $G$ being CAT(0), cocompactly cubulated or virtually special (Rips)

Definition

$G$ is Special group if $G={\pi }_1(X)$ for a compact special cube complex. (Or if it is a comxev-cocompact subgroup of a Right-angled Artin group)

Any fundamental group of a $3$-mfd is special. Also they contain all of the easy examples above.

Theorem (F'21)

$G$ special and $Out(G)$ infinite, then $G$ splits over a Centraliser.

Theorem (F.'25)

$G$ is special and $1$-ended. Then $G$ splits as an $Out(G)$-invariant Graph of groups s.t.

• Edge groups are either centralisers or $\cong \mathbb{Z}$
• Product subgroups of $G$ are conjugate into vertex groups
• Vertex groups are either (rigid+convex cocompact or quadratically hanging)

Moreover, let $V$ be a rigid vertex group. Then $Out(V)$ virtually embeds into $Out(H_1)\times ...\times Out(H_k)$ for $H_k$ special groups of “lower complexity”.

Corollary

For any $G$ virtually special.

1. $Out(G)$ has finite virtual cohomological dimension
2. $Out(G)$ satisfies the alternative
   • any subgroup is virtually polycyclic
   • any subgroup contains $F_2$

• What is the constant issue with torsion?