Totally geodesic surfaces with arbitrarily many compressions

TL;DR

This paper proves that for certain surfaces in the figure eight knot complement, the number of Dehn fillings making them compressible can be arbitrarily large, answering a question about their behavior.

Contribution

It demonstrates that there is no universal upper bound on the number of compressible Dehn fillings for totally geodesic surfaces, independent of the surface.

Findings

No universal upper bound on compressible fillings for these surfaces

Finitely many fillings keep the surface incompressible

Answers a longstanding question of Ying-Qing Wu

Abstract

A closed totally geodesic surface in the figure eight knot complement remains incompressible in all but finitely many Dehn fillings. In this paper, we show that there is no universal upper bound on the number of such fillings, independent of the surface. This answers a question of Ying-Qing Wu.