Riemann Surfaces of Genus 9 with Nilpotent Automorphism Group of the Largest Order Are Not Unique: On a Question in Farb's Problem 2.19

Problem 2.19 of Farb's list of problems on mapping class groups asks for properties of automorphism groups that determine a unique point of the moduli space \(\mathcal M_g\), and asks as an example whether there is a unique Riemann surface of genus \(g\ge2\) whose automorphism group is nilpotent and of the largest order possible for a nilpotent group of automorphisms of a surface of genus \(g\). By a theorem of Zomorrodian this largest order is \(16(g-1)\), attained only when \(g-1\) is a power of two. We show that the surface is not unique: there are at least four pairwise non-isomorphic compact Riemann surfaces of genus \(9\) whose full automorphism group is a \(2\)-group of order \(128=16(g-1)\). The surfaces are quotients of the hyperbolic plane by torsion-free normal subgroups of the triangle group \(\Delta(2,4,8)\), given by explicit presentations of the four groups of order \(128\), and the proof that each of them is the full automorphism group uses, besides finite computations, only that the triangle group \(\Delta(2,3,8)\) is the only Fuchsian group properly containing \(\Delta(2,4,8)\), which we prove. The presentations are taken from Conder's list of large group actions, and all verifications are finite computations in groups of order \(128\) and \(3072\), carried out with programs written for this note. Under the completeness of Conder's list, which two independent enumerations confirm for these genera, the number of such surfaces is \(1\), \(1\), \(4\), \(3\), \(7\) and \(15\) for \(g=3,5,9,17,33,65\), and none for \(g=2\) under our reading of the question. The four groups of genus \(9\) are also listed, with their generating vectors, in the database LMFDB. The first sentence of Problem 2.19 is open-ended and is not addressed. No statement of the result was found in the literature accessible to us; a search that finds nothing is not a proof of novelty. This is an unrefereed note.