Talk:Fundamental groups of surfaces (Ex)

Let $S$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}S$ be a surface. Suppose $\chi(S) > 0$$\chi(S) > 0$. Then $\pi_1(S)$$\pi_1(S)$ is finite and hence hyperbolic.

Suppose $\chi(S) < 0$$\chi(S) < 0$. Then the interior of $S$$S$ is a quotient of $\mathbb{H}^2$$\mathbb{H}^2$ by a discrete torsion-free subgroup $\Gamma$$\Gamma$ of $PSL_2(\mathbb{R})$$PSL_2(\mathbb{R})$. Then $\pi_1(S)$$\pi_1(S)$ is hyperbolic since it is isomorphic to $\Gamma$$\Gamma$ which acts freely and properly discontinuously on $\mathbb{H}^2$$\mathbb{H}^2$.

Now let $\chi(S)=0$$\chi(S)=0$. If $S$$S$ has boundary, then it is homeomorphic to an annulus or a Mobius band. Each of which have fundamental group $\mathbb{Z}$$\mathbb{Z}$ which is hyperbolic since its Cayley graph is quasi-isometric to a tree.

The only remaining cases are the Klein bottle and the torus. The latter has fundamental group $\mathbb{Z}^2$$\mathbb{Z}^2$, while the fundamental group of the former contains $\mathbb{Z}^2$$\mathbb{Z}^2$ as an index 2 subgroup. Thus neither of these groups is hyperbolic.