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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:

[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

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  • $\begingroup$ I do not understand what the issue is with applying the theorem from [1]: Assume that there is a counter-example and then arrive to a contradiction using [1] and [3]. $\endgroup$ Sep 3, 2021 at 16:41
  • $\begingroup$ Yes, that is my hope. But can there be a closed essential surface embedded in the punctured torus bundle that is boundary-compressible? Or that is non-orientable? $\endgroup$
    – wandersam
    Sep 6, 2021 at 8:34
  • $\begingroup$ And I am also unsure about switching between the manifold with torus boundary components and the (hyperbolic) interior. $\endgroup$
    – wandersam
    Sep 6, 2021 at 9:09

1 Answer 1

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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.

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  • $\begingroup$ What you wrote is not really relevant to the question since the general Bass' theorem is about very general subgroups (without discreteness assumptions, etc). $\endgroup$ Sep 3, 2021 at 16:39
  • $\begingroup$ Cases (a) and (b) belong to HNN extensions and free products with amalgamation. From [1], Thm. 1.5.3, the existence of an incompressible embedded surface follows. So Bass's Theorem indeed only uses Bass-Serre theory for the (abstract) group structure. $\endgroup$
    – wandersam
    Sep 6, 2021 at 8:48
  • $\begingroup$ @MoisheKohan (forgot to tag you) $\endgroup$
    – wandersam
    Sep 6, 2021 at 21:44

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