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$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Hom{Hom}$Let $S$ be a compact oriented surface with nonempty boundary. There are two variants of Teichmuller space for $S$ you might consider:

  1. The one that parameterizes finite-volume complete hyperbolic metrics on the interior of $S$. These correspond to discrete and faithful representations of the fundamental group of $S$ into $\PSL(2,\mathbb{R})$ that take the loops surrounding the punctures to parabolic elements.

  2. The one that parameterizes finite-volume complete hyperbolic metrics on $S$ with geodesic boundary. These correspond to (certain, not all as in 1) discrete and faithful representations of the fundamental group of $S$ into $\PSL(2,\mathbb{R})$ that take the loops surrounding the punctures to hyperbolic elements.

Let $U \subset \Hom(\pi_1(S),\PSL(2,\mathbb{R}))$ be the set of representations in either 1 or 2, so you obtain Teichmüller space from $U$ by quotienting out by the conjugation action of $\PSL(2,\mathbb{R})$.

Question: What is a good reference for the fact that $U$ is open? I know many good sources for the corresponding fact when $S$ is a closed oriented surface, where in fact we can replace $\PSL(2,\mathbb{R})$ by an arbitrary Lie group (a theorem of Weil — here we require the representation to be discrete, faithful, and cocompact). But I don't know a source that does these variants.

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    $\begingroup$ In case 1, $U$ is not open in the space of all representations as you can deform the parabolic to an elliptic with infinite order and get a non-discrete group. It is probably open in the subspace of type-preserving representations (those that send peripheral elements to parabolics), but I don't know a reference for that. $\endgroup$ Aug 4, 2020 at 18:46
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    $\begingroup$ In case 2 I also don't know a reference, but I think you can bootstrap an argument from the closed case: let $2S$ be the double along the boundary, all representations of $\pi_1(S)$ extend to representations of $\pi_1(2S)$ which are equivariant with respect to the involution given by the symmetry in the boundary. The map between representation spaces is an embedding and the result follows. (I did not check the details but they do not seem hard to write up.) $\endgroup$ Aug 4, 2020 at 18:47

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$\DeclareMathOperator\PSL{PSL}$Here is a complement to Jean Raimbault's first comment (I would have posted it as a comment, but I have not yet unlocked that privilege). Let $S$ be a thrice-punctured sphere, and let $\Gamma$ be the subgroup of $\PSL_2\mathbb{R}$ generated by $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$, $\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}$. We may identify $\Gamma$ with $\pi_1(S)$ and view the inclusion $\Gamma \hookrightarrow \PSL_2\mathbb{R}$ as a holonomy representation associated to the unique complete finite-area hyperbolic structure on $S$.

Here is a sequence of representations $\rho_n : \Gamma \rightarrow \PSL_2\mathbb{R}$ converging to the inclusion $\Gamma \hookrightarrow \PSL_2\mathbb{R}$ such that each of the $\rho_n$ is discrete but not faithful. We work in the upper half-plane model of $\mathbb{H}^2$. For $n\geq 3$, let $L_n$, $L_n'$ be the geodesics in $\mathbb{H}^2$ of equal Euclidean length passing through the real points $-1$, $1$, respectively, and intersecting at a purely imaginary point $z_n$ such that the angle facing $\infty$ formed at $z_n$ by $L_n$, $L_n'$ is $2\pi/n$. Let $\gamma_n \in \PSL_2\mathbb{R}$ be the clockwise "rotation" by $2\pi/n$ fixing $z_n$ (so $\gamma_n$ maps $L_n$ onto $L_n'$). Define $\rho_n$ by $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \mapsto \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$, $\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix} \mapsto \gamma_n$. For each $n\geq 3$, the map $\rho_n$ is a holonomy representation $\pi_1(S) \rightarrow \PSL_2\mathbb{R}$ associated to an incomplete finite-area hyperbolic structure on $S$: namely, the holonomy representation associated to the thrice-punctured sphere obtained by removing the cone point from the orbifold $\rho_n(\Gamma)\backslash \mathbb{H}^2$.

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You know, I also have been looking for a reference for openness in the type preserving setting and haven't been able to find it. (Actually, I'd like to have a theorem like this that works for orbifolds, even.)

At worst, I think you can prove it using the same arguments as in Weil's paper, though. Namely, Weil's proof (phrased in the $\mathbb H^2$ case) essentially involves taking a huge compact subset $K\subset \mathbb H^2$ that includes a fundamental domain of the image of a discrete, faithful representation $\rho$, letting $\Delta \subset \pi_1 S$ be a finite subset that includes all $\gamma\in \pi_1 S$ such that $\rho(\gamma)$ translates $K$ anywhere near itself, and then for $\rho'\approx \rho$, showing that $K / \rho'(\Delta)$ is a compact hyperbolic $2$-orbifold with $\rho'$-holonomy, implying that $\rho'$ is discrete and faithful. If $\rho$ has parabolics, one can instead take $K$ large enough so that it projects to a compact core of the quotient. Then $K / \rho'(\Delta)$ will be an (incomplete) hyperbolic $2$-orbifold with $\rho'$-holonomy, but since $\rho'$ is type preserving, one can ensure that the ends of this incomplete orbifold have parabolic holonomy, and hence the orbifold is contained in a complete finite volume orbifold.

Haven't really thought through the details, though.

Edit: Ah, it also follows from Theorem 1.1 in Bergeron and Gelander - A note on local rigidity, for example.

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