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Automatic continuity of homomorphisms and fixed points on metric compacta

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Abstract

We 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Christian Rosendal & Sławomir Solecki

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  1. Christian Rosendal
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  2. Sławomir Solecki
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Correspondence to Christian Rosendal.

Additional information

Research of the second author was supported by NSF grant DMS-0400931.

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Rosendal, C., Solecki, S. Automatic continuity of homomorphisms and fixed points on metric compacta. Isr. J. Math. 162, 349–371 (2007). https://doi.org/10.1007/s11856-007-0102-y

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  • DOI: https://doi.org/10.1007/s11856-007-0102-y

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