Geometry & Topology, Vol. 8 (2004)
Paper no. 23, pages 877--924.
Global rigidity of solvable group actions on S^1
Lizzie Burslem, Amie Wilkinson
Abstract.
In this paper we find all solvable subgroups of Diff^omega(S^1) and
classify their actions. We also investigate the C^r local rigidity of
actions of the solvable Baumslag-Solitar groups on the circle.
The
investigation leads to two novel phenomena in the study of infinite
group actions on compact manifolds. We exhibit a finitely generated
group Gamma and a manifold M such that:
* Gamma has exactly
countably infinitely many effective real-analytic actions on M, up to
conjugacy in Diff^omega(M);
* every effective, real analytic
action of Gamma on M is C^r locally rigid, for some r>=3, and for
every such r, there are infinitely many nonconjugate, effective
real-analytic actions of Gamma on M that are C^r locally rigid, but
not C^(r-1) locally rigid.
Keywords.
Group action, solvable group, rigidity, Diff^omega(S^1)
AMS subject classification.
Primary: 58E40, 22F05.
Secondary: 20F16, 57M60.
DOI: 10.2140/gt.2004.8.877
E-print: arXiv:math.DS/0310498
Submitted to GT on 26 January 2004.
Paper accepted 28 May 2004.
Paper published 5 June 2004.
Notes on file formats
Lizzie Burslem, Amie Wilkinson
Department of Mathematics, University of Michigan
2074 East Hall, Ann Arbor, MI 48109-1109 USA
and
Department of Mathematics, Northwestern University
2033 Sheridan Road, Evanston, IL 60208-2730 USA
Email: burslem@umich.edu, wilkinso@math.northwestern.edu
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