Global fixed points for centralizers and Morita's Theorem

TL;DR

This paper establishes a fixed point theorem for centralizers of certain homeomorphisms, applies it to pseudo-Anosov maps, and provides a simplified proof of Morita's Theorem with an improved genus bound.

Contribution

It introduces a new fixed point theorem for centralizers with attractor-repeller dynamics and offers an elementary proof of Morita's Theorem, lowering the genus bound from 5 to 3.

Findings

Existence of global fixed points for centralizers with specific boundary dynamics

Finite index subgroups of centralizers of pseudo-Anosov maps have infinitely many fixed points

Simplified proof of Morita's Theorem with improved genus bound

Abstract

We prove a global fixed point theorem for the centralizer of a homeomorphism of the two dimensional disk $D$ that has attractor-repeller dynamics on the boundary with at least two attractors and two repellers. As one application, we show that there is a finite index subgroup of the centralizer of a pseudo-Anosov homeomorphism with infinitely many global fixed points. As another application we give an elementary proof of Morita's Theorem, that the mapping class group of a closed surface $S$ of genus $g$ does not lift to the group of diffeormorphisms of $S$ and we improve the lower bound for $g$ from 5 to 3.