The Hurwitz automorphisms problem for translation surfaces

In 1893, Hurwitz proved that the group of automorphisms of a Riemann surface is finite and has order at most 84(g-1). This bound is reached for certain genera (3, 7, 14, 17, 118, 129 ... see OEIS entry A179982), but it is generally difficult to calculate the maximum number of automorphisms that a Riemann surface of a given genus g can have. This problem is known as the Hurwitz automorphisms Problem, and despite several results (starting with those of Accola and Maclachlan, who independently obtained the lower bound 8(g+1) in 1968–1969, which holds for an infinite family of genera), it remains largely open.

In this talk, we focus on a variant of the problem concerning translation surfaces. A translation surface can be viewed as a Riemann surface X equipped with a holomorphic 1-form ω, and the corresponding automorphism group Aut(X,ω) is therefore a subgroup of the group Aut(X) of automorphisms of the corresponding Riemann surface. J.C. Schlage-Puchta and G. Weitze-Schmidthüsen demonstrated in 2017 that the order of this group cannot exceed 4(g-1), and characterized the genera for which this bound is achieved. In a joint work with R. Gutiérrez-Romo and E. Lanneau, we study the corresponding Hurwitz problem and show, for example, that the lower bound is 2(g-1), and that it is attained for an infinite family of genera of the form g = pq + 1, where p and q are distinct primes greater than 3. The aim of this presentation is to introduce the main concepts and ideas.