Radhika Gupta - Introduction to free-by-cyclic groups - IV 3-Manifolds

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.

Lecture Notes: Muthanguha on his webpage.

3-Manifolds

Theorem (Stallings, 1961)

Let $M$ be compact irreducible 3-manifold. Given an element of the cohomology fulfilling some properties. Then $M$ fibers over $S^1$ with fiber a compact suface $\Sigma$ whose Fundamental group is given by a kernel.

The main take away is likely that we can correspond cohomology to fibrations.

Thurston fibered face theory Thurston gave a way of organising elements of the first cohomologu corresponding to fibrations. He

1. Defined a norm on first cohomology whose norm ball is a finite sided polyhedron
2. Primitive integral elements in the first cohomology lying over a given face are either all fibered or none.

In fact, there are finitely many faces that organise all the fibrations.

Theorem (Fried)

$M=M(\varphi )$, $\varphi :S_g\to S_g$ Pseudo-Anosov Homeomorphism. Take one element of the first cohomology. Let $F$ be the corresponding fibered face of the polyhedron. By Thurston’s hyp thm: Given a Primitive integral $\alpha$ we can construct a pseudo-Anosov ${\varphi }_{\alpha }:S_{\alpha }\to S_{\alpha }$.

Theorem (Fried)

There exists a convex, real analytic function $h:{\mathbb{R}}_{+}F\to \mathbb{R}$. such that

1. For all primitive integrals $\alpha \in {\mathbb{R}}_{+}F$ we have $h(\alpha )=\log \lambda (\alpha )$
2. constant on rays
3. goes to $\infty$ at $\partial {\mathbb{R}}_{t}F$

Theorem (McMullen)

Given a fibered face $F$. He defined a Teichmüller polynomial ${\Theta }_F\in \mathbb{Z}[H_1(M,\mathbb{Z})/torsion]$ which can be used to explicitly compute $\lambda \alpha$ for $\alpha \in {\mathbb{R}}_{+}\mathbb{F}$.

Definition (Bieri-Neumann-Strebel invariant '87)

Can be used for a systematic study of Algebraically fibered group.

Theorem

Let $G$ be a finitely presented group. Let $\Sigma (\Gamma )\subseteq H^1(G,\mathbb{R})$ be the BNS invariant

1. The kernel of a first cohomology is finitely generated iff. it lies at the boundary of the BNS invariant (?)
2. Let $u$ be an integral coefficient element from the BNS invariant. Then $G$ is an ascending HNN-Erweiterung over a f.g. base group $B\subseteq ker(U)$

Fibered face theory for free-by-cyclic group

This is a heavily studied area: Dowdall, Kapovich, Leininger, AlgomKfir, Hironaka, Rafi, Cashen, Levitt.

Theorem (DKL)

Let $\varphi$ be irreducible, $\infty$-order and expanding (Atoroidal monodromy). Let $G=F⋊\mathbb{Z}$. There is an open, convex rationally defined cone $C_0\subseteq H^1(G,\mathbb{R})$ containing a special map $u_0$ s.t. it is the component of $\Sigma (G)$ containing $u_0$.

And Kielak showed that there are finitely many components in $\Sigma (G)$.

1. And every primitive integral $u$ inside that component cprresponds to an asc HNN extension. In particular, if the $ker(u)$ is f.g. then $G=ker(u){⋊}_{{\varphi }_n}\mathbb{Z}$, ${\varphi }_n$ is also atoroidal.
2. They generalised Fried’s theorem
3. There is a McMullen polynomial

Invariants

• DKL invariant: Irreduducible atoroidal is inv. over a cone

• Dowdall-G-Taylor: $\varphi$ irred+ator

• openquestion List of open problems: http://aimpl.org/freebycyclic/