Finite translation surfaces with maximal number of translations (Q522290)

The authors consider translation surfaces, i.e., Riemann surfaces \(X\) with a finite covering \(p:X \to E\) of an elliptic curve \(E\) ramified above only one point, and they prove an analog to the famous Hurwitz bound for the number of automorphisms of compact Riemann surfaces: the number of automorphisms induced by translations in the charts is bounded from above by \(4g-4\) for all genera \(g>1\). This upper bound is obtained precisely if \(p\) is a normal origami cover and all ramifications are of order \(2\). So far, the result is a comparatively easy consequence of the Riemann-Hurwitz formula. Much more difficult are some facts about the structure of the possible maximal translation automorphism groups; they are based on some deep group theory. These groups are generated by two elements whose commutator is of order \(2\), and precisely all multiples of \(8\) and \(12\) occur as group orders. As a consequence, the possible genera are precisely all odd \(g>1\) and all \(g \equiv 1 \bmod 3\). The paper is very well written -- a pleasure to read!