Reference for openness of subspace of PSL(2,R) representation variety corresponding to Teichmüller space

Question

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Hom{Hom}$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.

Answer 1

$\DeclareMathOperator\PSL{PSL}$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 $\rho_n(\Gamma)\backslash \mathbb{H}^2$.

Answer 2

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.