Geodesic
Bounded orbits and global fixed points for groups acting on the plane
Mann, Kathryn1
1Department of Mathematics, University of Chicago, 5734 University Ave, Chicago IL 60637, USA
Algebraic and Geometric Topology, Tome 12 (2012) no. 1, pp. 421-433
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Let G be a group acting on ℝ2 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 > (1∕3)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 ℝ2 with uniformly bounded orbits has a global fixed point. The constant (1∕3)k is sharp.

As an application, we also show that a group acting on ℝ2 by diffeomorphisms with orbits bounded as above is left orderable.

DOI : 10.2140/agt.2012.12.421
Classification : 37E30, 57M60
Keywords: fixed point, planar action, group action, prime end, left order, plane homeomorphism, Brouwer plane translation

Mann, Kathryn  1

1 Department of Mathematics, University of Chicago, 5734 University Ave, Chicago IL 60637, USA 
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Mann, Kathryn. Bounded orbits and global fixed points for groups acting on the plane. Algebraic and Geometric Topology, Tome 12 (2012) no. 1, pp. 421-433. doi: 10.2140/agt.2012.12.421

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