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 Teichmuller 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.

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