I am reading MacLachlan / Reid: The Arithmetic of Hyperbolic 3-Manifolds.
Bass's Theorem (Thm. 5.2.2) states that a finite volume hyperbolic 3-manifold $M = H^3 / \Gamma$, with $\Gamma \subset SL(2, \mathbb{C})$ discrete, that has non-integral trace (i. e. $\mathrm{tr}\, g$ is not an algebraic integer for some $g \in \Gamma$) contains a closed embedded essential surface.
Definitions
The book does not give a definition of essentiality but I believe it is defined as in Cooper / Long / Reid: Essential Closed Surfaces in Bounded 3-Manifolds, Introduction: an embedding $i: S \to M$ of a closed, orientable connected surface $S$ is called essential if $\pi_1 i: \pi_1 S \to \pi_1 M$ is injective and $(\pi_1 i)(\pi_1 S)$ cannot be conjugated into a subgroup $\pi_1 (\partial_0 M)$ of $\pi_1 M$ where $\partial_0 M$ is a component of $\partial M$.
In section 1.5.1, embeddings $f: S \to M$ are defined to require $f(\partial S) \subseteq \partial M$. An embedded surface $f: S \to M$ is called compressible if $S$ is a 2-sphere and $f(S)$ bounds a 3-ball in $M$ or if $\pi_1 f: \pi_1 S \to\pi_1 M$ is not injective. Otherwise it is called incompressible.
The proof
In sections 5.2.1 and 5.2.2, the authors give a proof which I am able to follow up to the point where the existence of an embedded incompressible surface which is not boundary-parallel is deduced.
In the last sentence (p. 170) the proof then goes on to state that according to Thm. 1.5.3 , the surface can also be chosen to be closed. Closedness is neither mentioned in the formulation of this theorem (nor anything I can connect to it) nor in its sibling Thm. 1.5.2:
Leaving out the assumptions of Thm. 1.5.2 and 1.5.3 which are conditions on the algebraic structure of the fundamental group, the theorems say
Then $M$ contains an embedded incompressible surface that is not boundary parallel. (Thm. 1.5.2)
and
Then $M$ contains an embedded incompressible surface. Furthermore if $C$ is a connected subset of $\partial M$ for which the image of $\pi_1 C$ in $\pi_1 M$ is contained in a vertex stabilizer, then the surface may be taken disjoint from $C$. (Thm. 1.5.3)
respectively. The term "vertex stabilizer" is to be understood in the context of Bass-Serre Theory again (with $\pi_1 M$ acting on the tree of $SL(2, K_P)$, see section 5.2.1).
Do you have an idea how to fill this gap?
You could try to simply take the closure of the surface within $M$, but the problem is that a cusped hyperbolic manifold $M$ will only be the interior of a compact manifold with torus boundary components. Thm. 1.5.3 ensures that the surface can be taken disjoint from a boundary component, but I do not see why this extends to the closure of the surface.
(I am also tagging group-theory in case this is a consequence of the Bass-Serre theory involved in the proof.)
Related questions discussing the intended application of Bass's Theorem:
Given an embedded incompressible surface $S$ in $M$, the requirement that $\partial S$ be embedded in $\partial M$ can be coupled with conclusion of Theorem 1.5.3 which gives conditions on a component $C$ of $\partial M$ which ensure that $S$ is disjoint from that component. The condition says that the image of $\pi_1 C$ in $\pi_1 M$ is contained in a vertex stabilizer.
So, suppose we know the following two properties:
It follows that $S$ is disjoint from every component of $\partial M$, and therefore $\partial S \subset S \cap \partial M = \emptyset$. Therefore we may conclude that $S$ is closed. Of course, you have to check that property 1 is actually true in order to reach that conclusion.