Cofinal elements and fractional Dehn twist coefficients

TL;DR

This paper proves that certain Dehn twists are cofinal in all left orderings of the mapping class group of a surface with positive genus and one boundary, and shows the invariance of fractional Dehn twist coefficients across different actions.

Contribution

It establishes the cofinality of boundary-parallel Dehn twists in all left orderings and demonstrates the independence of fractional Dehn twist coefficients from the choice of action.

Findings

Dehn twists along boundary-parallel curves are cofinal in all left orderings.

Fractional Dehn twist coefficients are invariant under different group actions.

Provides a formula to recover fractional Dehn twist coefficients from any left ordering.

Abstract

We show that for a surface $S$ with positive genus and one boundary component, the mapping class of a Dehn twist along a curve parallel to the boundary is cofinal in every left ordering of the mapping class group $\operatorname{Mod}(S)$. We apply this result to show that one of the usual definitions of the fractional Dehn twist coefficient -- via translation numbers of a particular action of $\operatorname{Mod}(S)$ on $\mathbb{R}$ -- is in fact independent of the underlying action when $S$ has genus larger than one. As an algebraic counterpart to this, we provide a formula that recovers the fractional Dehn twist coefficient of a homeomorphism of $S$ from an arbitrary left ordering of $\operatorname{Mod}(S)$.