Talk:Extensions of groups (Ex)

Suppose that $G$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}G$ is the fundamental group of a closed irreducible $3$$3$-manifold $M$$M$, and that we have the exact sequence $1 \to H \to G \to \mathbb{Z} \to 1$$1 \to H \to G \to \mathbb{Z} \to 1$. If $H$$H$ is finitely generated, then Stallings' theorem says that $H$$H$ is the fundamental group of a closed surface $F$$F$, and moreover that $M$$M$ can be realised as a fibre bundle over $S^1$$S^1$ where $F$$F$ is the fibre.

It is a theorem of Thurston that if $F$$F$ is a closed surface of genus $g\geq 2$$g\geq 2$, and $\rho:F\to F$$\rho:F\to F$ is a pseudo-Anosov diffeomorphism, then the mapping torus of $\rho$$\rho$ is a closed hyperbolic $3$$3$-manifold. Hence $G\cong \pi_1(M)$$G\cong \pi_1(M)$ is hyperbolic. Furthermore $H\cong \pi_1(F)$$H\cong \pi_1(F)$ is non-trivial, torsion-free, and finitely generated.