TY - JOUR
T1 - Automatic continuity of homomorphisms and fixed points on metric compacta
AU - Rosendal, Christian
AU - Solecki, Sławomir
N1 - Funding Information:
* Research of the second author was supported by NSF grant DMS-0400931. Received October 19, 2005 and in revised form February 14, 2006
PY - 2007/12
Y1 - 2007/12
N2 - We prove that arbitrary homomorphisms from one of the groups Homeo(2 ℕ ), Homeo(2ℕ)ℕ , Aut (ℚ, <), Homeo(ℝ}) or 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 of V. G. Pestov, that any action of the discrete group Homeo+(ℝ) by homeomorphisms on a compact metric space has a fixed point.
AB - We prove that arbitrary homomorphisms from one of the groups Homeo(2 ℕ ), Homeo(2ℕ)ℕ , Aut (ℚ, <), Homeo(ℝ}) or 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 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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U2 - 10.1007/s11856-007-0102-y
DO - 10.1007/s11856-007-0102-y
M3 - Article
AN - SCOPUS:57749188442
SN - 0021-2172
VL - 162
SP - 349
EP - 371
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
ER -