On Loops Intersecting at Most Once

  • Joshua Evan GreeneEmail author


We prove that on a closed, orientable surface of genus g, the maximum cardinality of a set of simple loops with the property that no two are homotopic or intersect in more than k points grows as a function of g like \(g^{k+1}\), up to a factor of \(\log g\). The proof of the upper bound uses arguments from probabilistic combinatorics and a theorem of Scott related to the fact that surface groups are LERF.

Mathematics Subject Classification

57M15 05C62 05D40 



I thank Ravi Boppana, Jacob Caudell, Jonah Gaster, and Larry Guth for fun and helpful conversations. In particular, Larry laid the groundwork for constructing the “enemy graphs" of Theorem 8. I thank Dan Margalit for drawing my attention to [SS00] and the referee for unpacking the proof of Lemma 10. This work was supported by NSF CAREER Award DMS-1455132.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsBoston CollegeChestnut HillUSA

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