Global fixed points for centralizers and Morita's Theorem
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2008
- DOI:
- 10.48550/arXiv.0801.0736
- arXiv:
- arXiv:0801.0736
- Bibcode:
- 2008arXiv0801.0736F
- Keywords:
-
- Mathematics - Dynamical Systems;
- Mathematics - Geometric Topology;
- 37E30;
- 57M60;
- 37C25
- E-Print:
- Geom. Topol. 13 (2009) 87-98