Fuchsian model

Let ${\displaystyle R}$ be a closed hyperbolic surface and let ${\displaystyle \Gamma }$ be a Fuchsian group so that ${\displaystyle \Gamma \backslash \mathbb {H} }$ is a Fuchsian model for ${\displaystyle R}$. Let $${\displaystyle A(\Gamma )=\{\rho \colon \Gamma \to \mathrm {PSL} _{2}(\mathbb {R} )\colon \rho {\text{ is faithful and discrete }}\}}$$ and endow this set with the topology of pointwise convergence (sometimes called "algebraic convergence"). In this particular case this topology can most easily be defined as follows: the group ${\displaystyle \Gamma }$ is finitely generated since it is isomorphic to the fundamental group of ${\displaystyle R}$. Let ${\displaystyle g_{1},\ldots ,g_{r}}$ be a generating set: then any ${\displaystyle \rho \in A(\Gamma )}$ is determined by the elements ${\displaystyle \rho (g_{1}),\ldots ,\rho (g_{r})}$ and so we can identify ${\displaystyle A(\Gamma )}$ with a subset of ${\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )^{r}}$ by the map ${\displaystyle \rho \mapsto (\rho (g_{1}),\ldots ,\rho (g_{r}))}$. Then we give it the subspace topology.

The Nielsen isomorphism theorem (this is not standard terminology and this result is not directly related to the Dehn–Nielsen theorem) then has the following statement:

For any ${\displaystyle \rho \in A(\Gamma )}$ there exists a self-homeomorphism (in fact a quasiconformal map) ${\displaystyle h}$ of the upper half-plane ${\displaystyle \mathbb {H} }$ such that ${\displaystyle h\circ \gamma \circ h^{-1}=\rho (\gamma )}$ for all ${\displaystyle \gamma \in \Gamma }$.

The proof is very simple: choose an homeomorphism ${\displaystyle R\to \rho (\Gamma )\backslash \mathbb {H} }$ and lift it to the hyperbolic plane. Taking a diffeomorphism yields quasi-conformal map since ${\displaystyle R}$ is compact.

This result can be seen as the equivalence between two models for Teichmüller space of ${\displaystyle R}$: the set of discrete faithful representations of the fundamental group ${\displaystyle \pi _{1}(R)}$ into ${\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )}$ modulo conjugacy and the set of marked Riemann surfaces ${\displaystyle (X,f)}$ where ${\displaystyle f\colon R\to X}$ is a quasiconformal homeomorphism modulo a natural equivalence relation.

• the Kleinian model, an analogous construction for 3-manifolds
• Fundamental polygon

Matsuzaki, K.; Taniguchi, M.: Hyperbolic manifolds and Kleinian groups. Oxford (1998).