A Finite Cover of the Genus-Two Surface in Which Every Four-Holed Sphere Has Connected Preimage: A Partial Answer to a Question of D. Gekhtman

In the problem session of the Oberwolfach workshop “New Trends in Teichmüller Theory and Mapping Class Groups” (2018), D. Gekhtman asked: given a finite cover \(f\colon S_h\to S_g\) of closed orientable surfaces with \(h>g\ge2\), is there always a four-holed sphere \(H\subset S_g\) with essential boundary components such that \(f^{-1}(H)\) is disconnected, and is there always such a three-holed sphere? We give a partial answer. For the four-holed question the answer is no in genus \(2\): there is a cover \(S_{10}\to S_2\) of degree \(9\), with monodromy group \(3^2{:}Q_8\) of order \(72\), such that the preimage of every four-holed sphere with essential boundary components is connected. The mechanism is a lifting obstruction: the monodromy homomorphism does not lift to the central extension \(3^{1+2}{:}Q_8\), whereas it would lift if some four-holed sphere had disconnected preimage. This settles the four-holed question in genus \(2\) only, where every such four-holed sphere is the complement of two disjoint curves, so that its boundary curves are isotopic in pairs; if the four boundary curves are required to be pairwise non-isotopic, there is no such subsurface in genus \(2\), the question begins in genus \(3\), and there it remains open. The example is not a regular cover, and its Galois closure is not an example. It belongs to a family of covers of \(S_2\) of degree \(p^2\) with group \(p^2{:}Q_8\), for every prime \(p\equiv3\pmod 4\); the same groups give no example in genus at least \(3\). In the positive direction we find, in every genus, three-holed and four-holed spheres with disconnected preimage for covers which factor through a non-trivial abelian cover, for abelian-by-cyclic monodromy groups, for regular covers with characteristic kernel, and when the genus is at least three times the order of the monodromy group. Exact computations in genus \(2\) show that \(9\) is the smallest degree of such an example, that in degrees \(9\) and \(10\) the only examples are the covers with group \(3^2{:}Q_8\) whose monodromy does not lift, and that there is none among the regular covers with deck group of order less than \(12180\). The three-holed question remains open (our covers do have pairs of pants with disconnected preimage), and so do the four-holed question in genus at least \(3\) and both questions for regular covers. This is an unrefereed note.