# Two definitions of Teichmüller space: relative isotopy or not?

The definition of Teichmüller space on wikipedia via marked Riemann surfaces say that two markings are equivalent if the map $$fg^{-1}$$ is isotopic to a holomorphic diffeomorphism.

The definition on John H. Hubbard's Teichmüller Theory And Applications To Geometry, however requires instead "isotopic relative to the ideal boundary $$I(S)$$". (It is actually proven in the book that "isotopy" may be replaced by "homotopy".)

Why are these two definitions equivalent?

"We said earlier that in most cases that interest us, the ideal boundary will be empty. So - for our purposes - the important condition of Teichmüller equivalence is that there exists an analytic isomorphism $$\alpha:X_1\rightarrow X_2$$ such that $$\varphi_2$$ is homotopic to $$\alpha\circ\varphi_1$$. In fact, some definitions of Teichmüller space omit any mention of the ideal boundary. But then Proposition 6.4.12 [crucial to the Bers embedding] is true only if the ideal boundary of the quasiconformal surface $$S$$ is empty."
If you read Chapter VI of Ahlfors' Lectures on Quasiconformal Mappings, you see that there is no mention of the ideal boundary in defining the Teichmüller space modeled on a surface $$S_0=\Bbb{H}/\Gamma_0$$. But then when it comes to constructing the Bers embedding, Ahlfors assumes that the Fuchsian group $$\Gamma_0$$ is of the first kind, i.e. in the action of $$\Gamma_0$$ on $$\partial\Bbb{H}=\Bbb{R}\cup\{\infty\}$$ every orbit is dense. This is equivalent to $$I\left(\Bbb{H}/\Gamma_0\right)=\emptyset$$.
It is also worthy to mention that the same type of confusion can come up in defining the mapping class group. Let $$S$$ be a quasiconformal surface; that is, a Riemann surface up to quasiconformal equivalence. With Hubbard's convention, in order to consider the moduli space $$\mathcal{M}_S$$ as the orbit space for the action of the mapping class group $${\rm{MCG}}(S)$$ on the Teichmüller space $$\mathcal{T}_S$$, the group $${\rm{MCG}}(S)$$ must be defined as the quotient $${\rm{QC}}(S)/{\rm{QC}}_0(S)$$ where $${\rm{QC}}(S)$$ is the group of quasiconformal self-homeomorphisms of $$S$$ and $${\rm{QC}}_0(S)$$ is the normal subgroup formed by elements which fix $$I(S)$$ and are isotopic to identity relative to it (Definition 6.4.13 in Hubbard's book). But then, in presence of ideal boundary, $${\rm{MCG}}(S)$$ is not discrete and $$\mathcal{T}_S$$ is infinite-dimensional (pp. 262 and 299 of the same book). That is because any quasisymmetric homeomorphism of $$I(S)$$ can be extended to a quasiconformal homeomorphism of $$S$$. But if in the definition of $${\rm{MCG}}(S)$$, one for instance works with quasiconformal homeomorphisms that are identity on $$I(S)$$, then the mapping class group may be discrete (e.g. the mapping class group of an open disk with finitely many punctures is a braid group). This latter convention of requiring homeomorphisms to be identity on the boundary is the one that appears on Wikipedia.