Geometry & Topology
- Geom. Topol.
- Volume 20, Number 5 (2016), 3033-3056.
Automatic continuity for homeomorphism groups and applications
Let be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of has the automatic continuity property: any homomorphism from to any separable group is necessarily continuous. This answers a question of C Rosendal. If is a submanifold, the group of homeomorphisms of that preserve also has this property.
Various applications of automatic continuity are discussed, including applications to the topology and structure of groups of germs of homeomorphisms. In an appendix with Frédéric Le Roux we also show, using related techniques, that the group of germs at a point of homeomorphisms of is strongly uniformly simple.
Geom. Topol., Volume 20, Number 5 (2016), 3033-3056.
Received: 18 August 2015
Revised: 3 February 2016
Accepted: 12 March 2016
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 54H15: Transformation groups and semigroups [See also 20M20, 22-XX, 57Sxx] 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms
Secondary: 03E15: Descriptive set theory [See also 28A05, 54H05]
Mann, Kathryn. Automatic continuity for homeomorphism groups and applications. Geom. Topol. 20 (2016), no. 5, 3033--3056. doi:10.2140/gt.2016.20.3033. https://projecteuclid.org/euclid.gt/1510859050