Automatic continuity of homomorphisms and fixed points on metric compacta

2006/04/26 by Christian Rosendal, Rosendal, Christian, Slawomir Solecki +1
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #math.GR #math.LO

paper · pdf · doi:10.48550/arxiv.math/0604575

arxiv created 2006/04/26 · arxiv updated 2009/12/01

Abstract

We prove that arbitrary homomorphisms from one of the groups \rm Homeo(\ca), \rm Homeo(\ca)^\N, \rm Aut(\Q,<), \rm Homeo(\R), or \rm Homeo(S¹) 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 on V.G. Pestov, that any action of the discrete group \rm Homeo₊(\R) by homeomorphisms on a compact metric space has a fixed point.

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