Commensurability between once-punctured torus groups and once-punctured Klein bottle groups (Q330679)

The fundamental groups of the once-punctured torus and Klein bottle are both free of rank \(2\), and so abstractly commensurable. In this case, they are also \textit{topologically commensurable}, in the sense that both surfaces are double-covered by the twice-punctured torus. On the other hand, if the two surfaces are endowed with a hyperbolic structure, one cannot expect them to have a finite common geometric cover in general. This can be restated by saying that if one considers a discrete and faithful representation in the isometry group of the hyperbolic plane of the once-punctured torus, so that the element representing the puncture is parabolic, and similarly for the once-punctured Klein bottle, the images of the two representations need not be commensurable (up to conjugacy) in the isometry group of the hyperbolic plane.NEWLINENEWLINEHere, the related question of understanding when faithful and type-preserving (i.e., where elements corresponding to cusps are mapped to parabolics) representations of the two groups can be commensurable in \(\mathrm{PSL}(2,\mathbb C)\) is considered. Note that the isometry group of the hyperbolic plane is naturally a subgroup of \(\mathrm{PSL}(2,\mathbb C)\).NEWLINENEWLINEMore precisely, the author provides a condition that ensures that, given a faithful and type preserving representation of the fundamental group of the once-punctured Klein bottle, there is a commensurable faithful and type-preserving representation of the fundamental group of the once-punctured torus.NEWLINENEWLINEA main step in the proof of the result is to show the following two facts: (1) Every type-preserving representation of the fundamental group of the once-punctured torus (resp. once-punctured Klein bottle) extends to a unique representation of an extension of the group corresponding to a double covering of an orbifold by the surface; moreover the representation of the group is faithful if and only if so is the representation of its extension. (2) The representations of the fundamental groups of the once-punctured torus and once-punctured Klein bottle are commensurable if and only if so are their extensions. The advantage of remarking these facts is that the extensions (i.e. fundamental groups of the quotient orbifolds) are generated by torsion elements: this makes their representations easier to construct and understand.NEWLINENEWLINESome considerations about Ford domains are also provided.