Vol. 317, No. 1, 2022

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Origamis associated to minimally intersecting filling pairs

Tarik Aougab, William Menasco and Mark Nieland

Vol. 317 (2022), No. 1, 1–20

Let Σn denote the symmetric group on n symbols and let σn denote the standard n-cycle (1,2,3,,n). We consider the combinatorial problem of counting n-cycles ρ so that the commutator [ρ,σn] is again an n-cycle. With a constructive recipe, we generate factorially many such n-cycles for odd n (when n is even, it is easy to see that no such n-cycles exist). We apply this to counting certain square-tiled surfaces where 2g 1 = n is the number of squares. Letting Sg denote the closed, orientable surface of genus g, in joint work with Huang, Aougab constructed exponentially many (in g) mapping class group orbits of pairs of simple closed curves whose complement is a single topological disk. Our construction produces factorially many (again in g) such orbits. These new orbits additionally have the property that the absolute value of the algebraic intersection number is equal to the geometric intersection number, implying that each pair naturally gives rise to an origami.

curves on surfaces, origamis, flat surfaces, mapping class groups
Mathematical Subject Classification
Primary: 57K20, 57M07, 57M15
Received: 24 August 2021
Revised: 18 February 2022
Accepted: 19 February 2022
Published: 19 June 2022
Tarik Aougab
Department of Mathematics and Statistics
Haverford College
Lancaster, PA
United States
William Menasco
Department of Mathematics
University at Buffalo
Buffalo, NY
United States
Mark Nieland
School of Mathematical Sciences
Rochester Institute of Technology
Rochester, NY
United States