The shape of a generic translation surface
Abstract
A translation structure equips a Riemann surface with a singular flat metric. Not much is known about the shape of a generic translation surface. We consider the stratum H_1(2g2) of translation surfaces of genus g with one singularity and show that the expected diameter of a surface is bounded above by a uniform multiple of ((log g)/g)^(1/2). This is smaller than what one would expect by analogy from the result of Mirzakhani about the expected diameter of a hyperbolic metric on a Riemann surface. In fact, more generally, we compute the expected value of the covering radius of a translation surface in any stratum H_1(kappa). To prove our result, we need an estimate for the volume of the thin part of H_1(kappa) which is given in the appendix.
 Publication:

arXiv eprints
 Pub Date:
 September 2018
 arXiv:
 arXiv:1809.10769
 Bibcode:
 2018arXiv180910769M
 Keywords:

 Mathematics  Geometric Topology;
 32G15;
 30F60;
 57M50;
 37P45
 EPrint:
 31 pages, 3 figures