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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 exampple.

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.

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-Gelander, for instance.

https://arxiv.org/pdf/1702.00342.pdf

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 exampple.

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.

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-Gelander, for instance.

https://arxiv.org/pdf/1702.00342.pdf