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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?

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I don't think they are the same. In fact, Hubbard points out that there exist different definitions on p. 255 of Teichmüller Theory And Applications To Geometry, Topology and Dynamics vol. 1:

"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.

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