Volume 12, issue 1 (2012)

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Bounded orbits and global fixed points for groups acting on the plane

Kathryn Mann

Algebraic & Geometric Topology 12 (2012) 421–433
Abstract

Let G be a group acting on 2 by orientation-preserving homeomorphisms. We show that a tight bound on orbits implies a global fixed point. Precisely, if for some k > 0 there is a ball of radius r > (13)k such that each point x in the ball satisfies g(x) h(x) k for all g,h G, and the action of G satisfies a nonwandering hypothesis, then the action has a global fixed point. In particular any group of measure-preserving, orientation-preserving homeomorphisms of 2 with uniformly bounded orbits has a global fixed point. The constant (13)k is sharp.

As an application, we also show that a group acting on 2 by diffeomorphisms with orbits bounded as above is left orderable.

Keywords
fixed point, planar action, group action, prime end, left order, plane homeomorphism, Brouwer plane translation
Mathematical Subject Classification 2010
Primary: 37E30, 57M60
References
Publication
Received: 11 November 2011
Accepted: 18 November 2011
Published: 20 March 2012
Authors
Kathryn Mann
Department of Mathematics
University of Chicago
5734 University Ave
Chicago IL 60637
USA
http://math.uchicago.edu/~mann