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Mathematics, Geometric Topology, Combinatorics, Differential Geometry, Mathematics, Number Theory
Abstract:
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.
