Radhika Gupta - Introduction to free-by-cyclic groups - I-III

Abstract

A Free-by-cyclic group is an extension of a Free group by an infinite cyclic group. This is a rich class of groups and has close connections to the study of automorphisms of free groups. The aim of this mini-course will be to give an introduction to free-by-cyclic groups and survey some important results in this area. In the first lecture, we will start with some basic properties of free-by-cyclic groups. In the second lecture, we will see when these groups are (relatively) hyperbolic and admit actions on CAT(0) cube complexes. In the third lecture, I will give an introduction to the ‘fibered face theory’ in analogy to Thurston’s theory for fibered 3-manifolds. Finally, we will conclude with some open questions about free-by-cyclic groups.

Talk

Definition

Definition

A Free-by-cyclic group is a semi-direct product of a free group and $\mathbb{Z}$ with presentation: $⟨a_1,...,a_n,t|t\alpha t^{-1}=\Phi (\alpha )⟩$ This is just a HNN-Erweiterung of a free group.

Lemma

If I have to automorphisms in the same Outer automorphisms of free group, then the Free-by-cyclic-groups are isomorphic.

There is a short exact sequence $1\to \mathbb{F}\to G\to \mathbb{Z}\to 1$

Definition

A free by cyclic group is a group extension of a free group by an $\infty$-cyclic groups. I.e. it is characterised by the above Kurze Exakte Sequenz.

But how do we extract the outer homomorphism out of the second definition? Given $1\to \mathbb{F}\to G\to \mathbb{Z}\to 1$ let $\tilde{z}$ be a lift of the generator $z$ of $\mathbb{Z}$ in $G$. Then there exists a hommomorphism $\chi :\mathbb{Z}\to Out(\mathbb{F})$ which sends $z$ to the automorphism $a\mapsto \tilde{z}a{\tilde{z}}^{-1}$. We can show that any two lifts only differ by conjugation, meaning they are in the same outer automorphism class. We also use the Splitting lemma somewhere.

Definition (Geometric)

Let $X$ be the wedge of circle on $n$ petals. Every automorphism $\varphi$ can be represented by a homotopy equivalence $f;X\to X$ st. $f_{\ast }=\varphi$. (Just think of the homotopy equivalences as petals bein sent to paths, modulo conjugation) We define the Mapping torus for a homotopy equivalence. The fundamental group of this mapping torus is given by ${\pi }_1(M(f))=⟨a_1,...,a_n,t|t^{-1}\alpha t=f(\alpha )⟩$ This is a free-by-cyclic group.

• What can we use to understand MCGs

We describe the relationship to HNN-Erweiterung.

Example

$\mathbb{F}{⋊}_{{\varphi }^n}\mathbb{Z}$ is a finite index subgroup of $\mathbb{F}{⋊}_{\varphi }\mathbb{Z}$.

Example

Let ${\mathbb{F}}^{\prime }$ be a finite index subgroup of $\mathbb{F}$.

1. If ${\mathbb{F}}^{\prime }$ is $\varphi$-inv. ${\mathbb{F}}^{\prime }{⋊}_{\varphi }\mathbb{Z}$ is a finite index subgroup of $\mathbb{F}{⋊}_{\varphi }\mathbb{Z}$
2. If its not $\varphi$-inv. then there are finitely many f.i. subgroups which are preserved by $\varphi$.

Example

$\mathbb{F}⋊\mathbb{Z}$ is Residually finite group.

This is a theorem right? Because we just present a proof to this example.

Some properties

• some can not act on CAT(0)-spaces
• some can act on CAT(0)-spaces
• Some are hyperbolic, they are characterised too
• All are coherent
• Some are Character hyperbolic group
• Solvable conjugacy problem
• Some researchers studied fibered face theory
• Hagen-Wise: If they are hyperbolic thay are cubilible
• Clay: Studied $l^{(2)}$-torsion
• Characterised irreducibles (?)
• Characterised character relative hyperbolic.
• Hughes-Kudlinska: Studied profinite rigidity
• Andrew-Martino: Studied the Automorphisms
• Kudlinska-Vaulina: They are equationally Noetherian
• Kielak-Linton: All one-relator groups with torsion are virtually free-by-cyclic.

Motivation by 3-manifolds: We start with a closed surface with negative euler characteristic. We define the Mapping torus. If we add a puncture, the Spline is homotopic to a bouquet of flowers and the picture is reduced to the study of free-by-cyclic groups.

Hyperbolicity

Theorem (Thurston '82)

The following are equivalent:

1. The mapping torus is a hyperbolic $3$-manifold
2. The fundamental group of the mapping torus does not contain ${\mathbb{Z}}^2$
3. The monodromy $\varphi$ is Atoroidal monodromy. (Meaning every simple closed curve is not mapps to a isotopic curve by a power of $\varphi$) i.e. it is homotopic to a Pseudo-Anosov Homeomorphism

Theorem (Brinkmann '02, Bestvina-Feighn-Handel)

The following are equivalent

1. $\mathbb{F}{⋊}_{\varphi }\mathbb{Z}$ is a Hyperbolic group
2. $\mathbb{F}{⋊}_{\varphi }\mathbb{Z}$ has no ${\mathbb{Z}}^2$ subgroup
3. $\varphi$ is atoroidal (${\varphi }^n(g)$ is not conj. to $g$)

Let $\varphi :\mathbb{F}\to \mathbb{F}$ be a injective endomoorphism.

Theorem (Mutanguha '21)

The following are equivalent:

1. $\mathbb{F}{\ast }_{\varphi }$ is a hyperbolic group
2. $\mathbb{F}{\ast }_{\varphi }$ has not Baumslag-Solitar Gruppen $BS(1,d)$ with $d\ge 1$
3. $\varphi$ is Strongly atoroidal group. (${\varphi }^n(g)=g^d$ are never conjugate)

There is a similar result for Relative hyperbolic group.

Theorem (Ghosh '23, Dahmani-Li '22)

A similar characterisation as above.

Open Problem: Are all hyperbolic free-by-cyclic groups quasi-isometric?

Actions on CAT(0) spaces

Theorem (Gerston '94)

Not all free-by-cyclic group can act geometrically on a CAT(0) space. We present one such example. This is a map where the curves grow linearly.

Hehe, Radhika gave a presentation. I can use my app to study this.

To show this we start with $\mathbb{Z}$ and attach HNN-extensions until we get our group. Since HNN-extensions interact nicely with CAT(0) space, this gives us some info.

Example (Igman '19, Samuelsen '06)

They constructed an example that does act on a CAT(0) space.

Theorem (Hagen-Wise '15, '16)

A hyperbolic free-by-cyclic group acts geometrically on a CAT(0) cube complex.

From this follows (Agol) that these groups are virtually Special group which means they are linear.

Theorem (Dahmani-Meda Satish0myutanguha '25)

Let $G$ and $G{⋊}_{\varphi }\mathbb{Z}$ hyperbolic. Then $G{⋊}_{\varphi }\mathbb{Z}$ is CAT(0).

Open Problem:

• Characterise free-by-cyclic acting on CAT(0) spaces or even CAT(0) cube complexes.
• Does every CAT(0) group act on a CAT(0) cube complex?
• If $\varphi$ is quadratically growing then is $\mathbb{F}{⋊}_{\varphi }\mathbb{Z}$ CAT(0)?
• Are $\mathbb{F}{⋊}_{\varphi }\mathbb{Z}$ injective? Ich sollte eine Sammlung von offenen Problemen erstellen, damit ich welche habe, wenn ich meinen PhD beginne.

Actions of cyclic splittings

Definition

A Free splitting is a group action $G$ on a tree $T$ with trivial edge stabiliser.

Example

Look at $F_2$ acting on the $4$-valent tree by the usual action.

Definition

A Cyclic splitting is a group action on a tree with a cyclic edge stabiliser.

As an example we mention the fundamental surface of $S_g$. This acts on tree consisting of the dual to the Dirichlet tessellation.

Proposition

Let $G$ a free-by-cyclic. It stabilises a free splitting of $F$ iff. $G$ admits a cyclic splitting.

In short, there is a connection to Bass-Serre tree One part of this is to extend the action of my free group on the tree to the free-by-cyclic group on the tree.

Fibered face theorem

We draw some pictures. We draw a surface/rose multiply by the unit interval, glue top and bottom and perform Surgery (cut open at some places and connect).

Next we draw a $3$-sheeted Cover of a $2$-petal rose. Its fundemental region then is a Finite index subgroup of $F_2$.

Radhika is really giving her best, but I can’t focus :/

Ok, so ${\pi }_1(S_g)\times \mathbb{Z}$ is obtained as the Fundamental group of a circle times $S_g$. We can show that by performing Dehn surgery, we can Connect multiple instances of $S_g$ (which are fibres of the torus). By doing this, we get a twisted fibration of the torus over a higher genus surface: ${\pi }_1(S_{g^{\prime }}){⋊}_{\varphi }\mathbb{Z}$.

Upshot: A free-by-cyclic group (over surface groups) can be fibered in $\infty$ many ways. By doing the same procedure over a rose, we get many different description of one free-by-cyclic group.

Example

We look at an example on the $3$-rose with $f:a\mapsto a,b\mapsto b,c\mapsto a{b}^{-1}c$

Definition

$\varphi \in Out(F_n)$ is called irreducible if $\varphi$ does not preserve the conjugacy class of any free factor of $F$.

Theorem (Daudall-Kapovich-Leininger '15, '17)

Let $G=F{⋊}_{\varphi }\mathbb{Z}=F^{\prime }{⋊}_{\psi }\mathbb{Z}$ such that the two fibrations are ‘in the same cone’. Then $\varphi$ is irreducible and innfinite order iff $\psi$ is irreducible and infinite order.

There is a whole list of different theorems of the same kind.

Theorem (Mutanguha)

If two mapping tori have the same fundamental group then $\varphi$ is irred. + infinite order iff. $\psi$ is irred. + infinite order.

Right, now I get the structure of the talk. We have three running examples. It should even be right up my alley :/