Recall that a hyperbolic 3-manifold $H^3/\Gamma$, $\Gamma \subset SL(2, \mathbb{C})$ discrete, is said to have integral traces if for all $\gamma \in \Gamma$, $\mathrm{tr} \gamma$ is an algebraic integer. ([1], Def. 5.2.1).
According to Bass's Theorem ([1], Thm. 5.2.2), a finite volume hyperbolic 3-manifold with non-integral trace contains a closed embedded essential surface.
According to [2], 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 $\partial M$.
A once-punctured torus bundle is a hyperbolic manifold which is homeomorphic to a mapping cylinder $M_\phi$ for $\phi: T^2 \setminus \{p\} \to T^2 \setminus \{p\}$ a (hyperbolic) homeomorphism of the once-punctured torus $T^2 \setminus \{p\}$.
The article [3] classifies the connected, orientable, incompressible, $\partial$-incompressible surfaces in a once-punctured torus bundle $M_\phi$ (Theorem 1.1). The only closed surface in this list is the peripheral torus $\partial M_\phi$.
Does it follow from this that once-punctured torus bundles have integral traces? The essentiality requirement from Bass's Theorem seems to exclude the peripheral torus. Wikipedia [4] states that (for proper embeddings) essentiality / algebraic incompressibility implies incompressibility. On the other hand, I do not know what to make of boundary-incompressibility and have no knowledge of the subtleties of low-dimensional geometry.
Related questions:
- The same question on Mathoverflow without discussing this specific attempt of proof: https://mathoverflow.net/q/403048/129446
- A discussion on the role of closedness in Bass's Theorem: Gap in Maclachlan/Reid's Proof of Bass's Theorem in Hyperbolic Geometry: Closedness?
[1] MacLachlan / Reid: The Arithmetic of Hyperbolic 3-Manifolds, section 5.2.
[2] Cooper / Long/ Reid: Essential Closed Surfaces in Bounded 3-Manifolds.
[3] Floyd / Hatcher: Incompressible Surfaces in Punctured-Torus Bundles.
[4] https://en.wikipedia.org/wiki/Incompressible_surface#Algebraically_incompressible_surfaces
It seems that Bass's Theorem cannot be used to prove that once-punctured torus bundles have integral traces:
The full version of the theorem [5] provides a "classification" of finitely generated subgroups of $GL(2, \mathbb{C})$ into four (non-exclusive) types. The types (c) and (d) imply integrality of traces. The holonomy groups of once-punctured torus bundles, however, are of type (a) (so we cannot rule out types (a) and (b) to deduce integrality of traces).
[5] Bass, Hyman: Chapter VI Finitely Generated Subgroups of Gl2. Pure and Applied Mathematics Vol. 112, 1984, pp 127-136.