Systolic pants decompositions with polynomial growth

Abstract

We investigate improved bounds on the kissing number of a hyperbolic surface under the assumption that the systole set contains pants decompositions with volume growth constraints. Namely, when the dual graph of the pants decomposition has polynomial volume growth, we obtain a power-savings improvement for the upper bound on the kissing number in terms of systolic length. We also construct super-exponentially many (in $g$) pants decompositions with exponential volume growth for which our techniques still imply stronger upper bounds for kissing numbers. The proofs use the technology of ``stem systems'' and ``tulips'' developed by the first and second author to study $1$-systems of curves.