Topology Seminar (Main Talk): Automorphisms of the k-curve graph

Seminar | September 19 | 4-5 p.m. | 3 Evans Hall

 Tarik Aougab, Brown University

 Department of Mathematics

The k-curve graph of an orientable surface S with negative Euler characteristic is a graph whose vertices correspond to (homotopy classes of) essential simple closed curves on S, and whose edges correspond to pairs of curves that geometrically intersect at most k times. For any surface with genus at least 3, we prove that the automorphism group of the 1-curve graph is isomorphic to the extended mapping class group; this resolves a conjecture of Schaller from 2000. More generally, we prove the same result for the k-curve graph so long as the absolute value of the Euler characteristic of S is at least 1000+k. This represents joint work with Yassin Chandran, Marissa Loving, Roberta Shapiro, Rob Oakley, and Sunny Yang Xiao.