The length of the shortest closed curve on a hyperbolic surface is bounded by a logartihmic function of its genus. It is also known that the length of the shortest closed curve on a congruence cover of an arithmetic surface is logarithmic in the genus of the cover. In this talk I will discuss a geometric construction of surfaces that have this property. This is joint work with Alexander Walker.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/6311