Automatic continuity of homomorphisms and fixed points on metric compacta

By Christian Rosendal, Slawomir Solecki. 2006

89 citations recorded by the source corpus.

AbstractWe prove that arbitrary homomorphisms from one of the groups $$Homeo(2^\mathbb{N} ), Homeo(2^\mathbb{N} )^\mathbb{N} , Aut(\mathbb{Q}, < ), Homeo(\mathbb{R}) or Homeo(S^1 )$$ into a separable group are automatically continuous. This has consequences for the representations of these groups as discrete groups. For example, it follows, in combination with a result of V. G. Pestov, that any action of the discrete group Homeo+(ℝ) by homeomorphisms on a compact metric space has a fixed point.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fb8954c738aec23c66f9fd20abec636546f9a979

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In Geometry and Topology

1. Javier Aramayona
2. Nicholas G. Vlamis
3. Priyam Patel

In Mathematical Physics

1. Javier Aramayona
2. Nicholas G. Vlamis
3. Priyam Patel

Citations in the corpus

• The First Integral Cohomology of Pure Mapping Class Groups (2017, methodology): “We note that descriptive set theory has been previously used in the study of homomorphisms of homeomorphism groups of compact manifolds, for instance see the work of Rosendal–Solecki [25], Rosendal [24], and Mann [21].”