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Abstract
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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.
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Keywords
homeomorphism groups, automatic continuity, germs of
homeomorphisms
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Mathematical Subject Classification 2010
Primary: 54H15, 57S05
Secondary: 03E15
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Publication
Received: 18 August 2015
Revised: 3 February 2016
Accepted: 12 March 2016
Published: 7 October 2016
Proposed: Leonid Polterovich
Seconded: Danny Calegari, Bruce Kleiner
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