Teichmüller spaces of Fuchsian groups

From now onwards, unless stated otherwise, we will assume that $ G$ is a finitely generated, torsion free Fuchsian group of the first kind, leaving the upper half plane invariant (the theory we explain below works in greater generality than this setting).

We define the space of bounded Beltrami coefficients for $ G$ on the upper half plane by

$\displaystyle L^\infty({\mathbb{H}},G) = \{\mu \in L^\infty({\mathbb{H}}) ; 
(\...
...line{g'}/g = \mu, \mathrm{a.e. on } {\mathbb{H}}, 
\mathrm{for all } g\in G\},
$

If $ \mu$ is in the unit ball of $ L^\infty({\mathbb{H}},G)$ (we will denote this unit ball by $ B(L^\infty({\mathbb{H}},G))$), let $ w_\mu$ denote the solution of the Beltrami equation, fixing $ \infty$, 0 and $ 1$ pointwise, obtained by extending $ \mu$ to the lower half plane by reflection. Let $ E_\mu:G\to G_\mu\subset \mathrm{PSL}(2,{\mathbb{C}})$ be the group homomorphism given by $ E_\mu(g) = w_\mu g (w_\mu)^{-1}$. Assume $ \mu$ and $ \nu$ are elements of $ B(L^\infty({\mathbb{H}},G))$ such that the groups $ G_\mu$ and $ G_\nu$ are equal; set $ X = {\mathbb{H}}/G$, $ Y={\mathbb{H}}/G_\mu={\mathbb{H}}/G_\nu$. We have functions $ f_\mu, f_\nu:X\to Y$ induced by $ w_\mu$ and $ w_\nu$ respectively.

Prop 2.11   The following are equivalent:
(i) $ f_\mu$ is homotopic to $ f_\nu$;
(ii) $ w_\mu\vert _{\mathbb{R}} = w_\nu\vert _{\mathbb{R}}$;
(iii) $ w_\mu g (w_\mu)^{-1} = w_\nu g (w_\nu)^{-1}$, for all $ g \in G$.

We define the Teichmüller $ T(G)$ and Riemann $ M(G)$ spaces of $ G$ as the quotients of the unit ball $ B(L^\infty({\mathbb{H}},G))$ by the relations $ \sim$ and $ \sim_R$, defined as follows:
$ \boldsymbol{\bullet}$ $ \mu \sim \nu$ if $ w_\mu\vert _{\mathbb{R}} = w_\nu\vert _{\mathbb{R}}$;
$ \boldsymbol{\bullet}$ $ \mu \sim_R \nu$ if $ G_\mu$ and $ G_\nu$ are conjugate subgroups of $ \mathrm{PSL}(2,{\mathbb{C}})$.

To define the modular group of $ G$ we need to set up some notation:

\begin{displaymath}
\begin{split}
Q(G) = & \{w:{\mathbb{H}} \to {\mathbb{H}};w \...
...y}\};\\
N_{qc}(G) = & \{w\in Q(G); wGw^{-1} = G\}.
\end{split}\end{displaymath}

We set $ mod(G) = N_{qc}(G) / \left(N_{qc}(G)G \cap Q_0(G)\right)$ and the modular group of $ G$ by $ Mod(G) = mod(G) / G$. If $ X = {\mathbb{H}}/G$ then the groups $ Mod(X)$ and $ Mod(G)$ are isomorphic (to obtain that $ Mod(G)$ is isomorphic to $ Mod(X)$, one has to quotient $ mod(G)$ by $ G$ since any homeomorphism of $ X$ has $ G$-lifts to $ \mathbb{H}$).

The space $ T(G)$ has a pseudo-metric $ \tau$ given by

$\displaystyle \tau([w_1],[w_2]) = \inf \frac{1}{2} \log K(w),
$

where $ w$ is a qc mapping, and $ w$ and $ w_1\circ w_2^{-1}$ coincide on the real axis.

Prop 2.12   The pseudo-metric $ \tau$ on $ T(G)$ is actually a metric. Moreover, if $ [\mu]$ and $ [\nu]$ in $ T(G)$ with $ \tau([\mu],[\nu]) = d$, then for any given $ \mu_0$ such that $ [\mu_0] = [\mu]$, there exists $ \nu_0$ such that $ [\nu_0] = [\nu]$ and $ \frac{1}{2}\log K(w_{\mu_0} \circ w_{\nu_0}^{-1}) = d$. $ \tau$ is a complete metric on $ T(G)$.

Prop 2.13   If $ G$ is a (non-elementary, torsion free) Fuchsian group and $ X = {\mathbb{H}}/G$, then there exists a canonical isometry between $ T(G)$ and $ T(X)$.

Proof. By conjugation we can assume that $ G$ is normalized; that is, $ \infty$, 0 and $ 1$ are points in the limit set of $ G$. Let $ \mu$ be an element of $ L^\infty({\mathbb{H}},G)$; then we assigned to it (the Teichmüller class of) the marked Riemann surface $ (X,f_\mu,X_\mu={\mathbb{H}}/G_\mu)$, where $ f_\mu$ is induced by $ w_\mu$ as above.

For the inverse mapping, given a marked surface $ (X,f,X_1)$, write $ X_1
= {\mathbb{H}}/G_1$; then $ f$ lifts to $ \tilde{f}:{\mathbb{H}} \to
{\mathbb{H}}$ such that $ \tilde{f}G\tilde{f}^{-1} = G_1$, and $ \tilde{f}$ fixes $ \infty$, 0 and $ 1$ pointwise. We assign to the given marked Riemann surface the Beltrami coefficient $ \mu_{\tilde{f}}$. $ \qedsymbol$

Theorem 2.14   The modular group $ Mod(G)$ ($ Mod(X)$) acts properly discontinuously by isometries on $ T(G)$ ($ T(X)$ respectively).

The proof of the theorem for $ G$ is based on the next two results. For a surface $ X$ one uses the isometries between $ T(G)$ and the identification between the groups $ Mod(G)$ and $ Mod(X)$.

Lemma 2.15   If $ G$ is Fuchsian, non-cyclic, then the normalizer of $ G$ in $ \mathrm{PSL}(2,{\mathbb{R}})$, $ N(G)$, is also Fuchsian. Moreover, if $ G$ is of the firs kind then $ N(G)/G$ is a finite group, isomorphic to the automorphism group of the surface $ {\mathbb{H}}/G$.

Lemma 2.16   If $ G$ is Fuchsian of the first kind then the set of traces squares of elements of $ G$ is a discrete subset of $ \mathbb{R}$.

Pablo Ares Gastesi 2005-08-31