Mathematical Research Letters

Volume 25 (2018)

Number 6

Graphs of large girth and surfaces of large systole

Pages: 1937 – 1956



Bram Petri (Mathematisches Institut, Universität Bonn, Germany)

Alexander Walker (Department of Mathematics, Brown University, Providence, Rhode Island, U.S.A.)


The systole of a hyperbolic surface is bounded by a logarithmic function of its genus. This bound is sharp, in that there exist sequences of surfaces with genera tending to infinity that attain logarithmically large systoles. These are constructed by taking congruence covers of arithmetic surfaces.

In this article we provide a new construction for a sequence of surfaces with systoles that grow logarithmically in their genera.We do this by combining a construction for graphs of large girth and a count of the number of $\mathrm{SL}_2 (\mathbb{Z})$ matrices with positive entries and bounded trace.

Received 3 February 2016

Accepted 24 August 2017

Published 25 March 2019