My question is about understanding of equivalence of two different definition for the Teichmüller space of hyperbolic surfaces $\mathbb{S}^2 \setminus P$, where $|P| \geq 3$.
The first definition for a general case (compact hyperbolic surface with finitely many interior punctures) can be found in B. Farb, D. Margalit A Primer on Mapping Class Group at the beginning of section 10.1, page 263-264.
Definition 1. The Teichmüller space $\mathrm{Teich(S)}$ of surface $S$ is a set of pairs $(X, \phi)$, where $X$ is a surface with a complete, finite-area hyperbolic metric with totally geodesic boundary and $\phi\colon S \to X$ is a diffeomorphism, quotient by a certain equivalence relation.
Two pairs $(X_1, \phi_1)$ and $(X_2, \phi_2)$ are equivalent if there is an isometry $I\colon X_1 \to X_2$ so that $I \circ \phi_1 \colon S \to X_2$ and $\phi_2\colon S \to X_2$ are homotopic. This is to say that the following
diagram commutes up to homotopy:
Also authors mention that "Here homotopies are allowed to move points in the boundary of $X_2$."
The second definition for a particular case of hyperbolic 2-sphere with punctures, i.e., for $\mathbb{S}^2 \setminus P$ with $|P| \geq 3$, is provided in the paper A proof of Thurston's topological characterization of rational functions by A. Douady and J.H. Hubbard on the tope of page 5.
Definition 2. The space $\mathcal{T}_p$ is the space of diffeomorphisms $\phi\colon \mathbb{S}^2 \to \mathbb{P}^1$ ($\mathbb{P}^1$ is the Riemann sphere, complex projective line) with $\phi_1$ and $\phi_2$ identified if and only if there exists an analytic isomorphism $h\colon \mathbb{P}^1 \to \mathbb{P}^1$ such that the diagram
commutes on $P$, and commutes up to isotopy rel $P$. This is to say that $\phi_2^{-1} \circ h \circ \phi_1: \mathbb{S}^2 \to \mathbb{S}^2$ is isotopic to the indentity rel $P$.
I am trying to understand why those definitions are really the same. My thoughts can be divided in the following 3 items:
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Going from $\mathcal{T}_P$ to $\mathrm{Teich}(\mathbb{S}^2 \setminus P)$. This seems to be the easiest one. Seems that we can simply restrict $\phi$'s and $h$'s onto $\mathbb{S}^2 \setminus P$ and end up with saying that complex structure on $\mathbb{P}^1 \setminus \phi(P)$ will give us unique hyperbolic metric with all necessary conditions. Unfortunately, I am a bit stuck with formulating the last argument precisely.
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Going from $\mathrm{Teich}(\mathbb{S}^2 \setminus P)$ to $\mathcal{T}_P$. Seems that here we have to somehow continue diffeomorphisms $\phi$'s and isometries $I$'s to the punctures $P$. This sounds like removable singularities theorem for holomorphic maps. But I do not know how to really apply it, because in it we want to have a punctured complex chart, but in our case punctures are outside of given complex structures. Also, up to homeomorphism we can do something weird with the punctures, for example, make from them holes obtained by removing closed disks. Probably this can be resolved by argument about finite volume, which implies that such a hyperbolic surface cannot contain such, kind of, trumpets, but just cusps. But still, how to do it precisely and why we can't have other weird behavior?
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Are the mapping class groups corresponding to those definitions the same? I guess that in the first case we have mapping class group $\mathrm{MCG}(\mathbb{S}^2 \setminus P)$ which can permute the punctures. In the second one we have pure mapping class group of sphere with marked points $\mathrm{PMCG}(\mathbb{S}^2, P)$, so all its elements are the identity on $P$. For example, in the case of $|P| = 3$ we will have $\mathrm{MCG}(\mathbb{S}^2 \setminus P) = S_3$ and $\mathrm{Teich}(\mathbb{S}^2 \setminus P)$ will contain exactly 6 points. From the other side, $\mathrm{PMCG}(\mathbb{S}^2, P) = \{\mathrm{Id}\}$ and $\mathcal{T}_P$ consists of the unique point. The second one looks more correct for me. Probably, I am missing something? I suppose that we want Teichmüller space to be connected (even simply connected), which is not the case above.
Best Answer
Suppose that $S$ is a compact connected oriented surface. Suppose that $P \subset S$ is a collection of marked points (none on the boundary). Let $g = g(S)$ be the genus of $S$. Let $p = |P|$ and let $b = |\partial S|$.
The space given in the Primer has real dimension $6g - 6 + 2p + 3b$. The space given in Douady-Hubbard has real dimension $-6 + 2p$.
There is no way to "turn boundary components into punctures" or conversely. This is because the the conformal structure of the interior of a surface with geodesic boundary "remembers" the length of the geodesic boundary.
If you disallow boundary and genus, then you are comparing spheres with marked points to spheres with punctures (with conformal structures and hyperbolic metrics, respectively).
(1) From a marked conformal sphere, the uniformisation theorem produces a unique hyperbolic metric on the punctured sphere.
(2) The Riemann removable singularities theorem produces the desired conformal structure by "continuing" the complex structure to the missing point.
(3) The mapping class groups are the same if you make the same decision in both cases about how homeomorphisms to permute marked points (respectively, permute the ends of the surface coming from punctures).