Hyperbolic punctured spheres without arithmetic systole maximizers

2022/09/05 by Lakeland, Grant S., Young, Clayton
#05C10 #57M15 #57M50 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2209.01748

Abstract

We find bounds for the length of the systole -- the shortest essential, non-peripheral closed curve -- for arithmetic punctured spheres with n cusps, for n=4 through n=12, some of which were previously known due to Schmutz. This is shown using a correspondence between such surfaces and planar triangulations. We show that for n=7,10,11, arithmetic surfaces do not achieve the maximal systole length.

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