Once-punctured torus bundles are a well-studied class of hyperbolic 3-manifolds, but unfortunately I have been unable to find out whether they always have integral traces (in the sense of [1], Def. 5.2.1). Is this already known?


In several dozens of examples that I have calculated with the help of Snappy or the approach in [2], this seems to be the case. Moreover, for bundles with monodromy $LR^n$ this seems to follow from the calculations in [2].


Bass's Theorem ([3]; [1] Thm. 5.2.2) sometimes helps in this situation, but I am not enough of a topologist to tell (and am inclined to say it does not). See my question (and attempt of answer) on MSE: <https://math.stackexchange.com/q/4237616/478924>


When calculating the trace field with the approach of [2], there is a finite set of algebraic solutions one of which belongs to the geometric structure of the manifold. It might be worthwhile to mention that in all my examples, also the other, non-geometric solutions define a (non-discrete) representation with integral traces.


**Related question** discussing closedness in Bass's Theorem: <https://math.stackexchange.com/q/4243973/478924>


[1] MacLachlan, Reid: The Arithmetic of Hyperbolic 3-Manifolds.


[2] Rehmann, Vinberg: On a Phenomenon Discovered By Heinz Helling. Transformation Groups 22, 259–265 (2017).


[3] Bass, Hyman: Chapter VI Finitely Generated Subgroups of Gl2. Pure and Applied Mathematics Vol. 112, 1984, pp 127-136.



Do once-punctured torus bundles have integral traces?
Floyd and Hatcher give a classification of essential surfaces in once-punctured torus bundles. In particular, there are no closed incompressible surfaces (other than the boundary torus), if there were they would persist in high parameter Dehn fillings, contradicting Corollary 1.2:


*Floyd, W.; Hatcher, Allen E.*, [**Incompressible surfaces in punctured-torus bundles**](http://dx.doi.org/10.1016/0166-8641(82)90035-9), Topology Appl. 13, 263-282 (1982). [ZBL0493.57004](https://zbmath.org/?q=an:0493.57004).


Then the version of Bass's theorem you cite (see Theorem 5.2.2 of the reference below aka [1] above) which shows that for hyperbolic manifolds with non-integral traces contain a closed incompressible surface. Together, these results confirm your computations more broadly. Hyperbolic once-punctured torus bundles have integral traces.


*Maclachlan, Colin; Reid, Alan W.*, The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics. 219. New York, NY: Springer. xiii, 463 p. (2003). [ZBL1025.57001](https://zbmath.org/?q=an:1025.57001).


