## Algebraic & Geometric Topology

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

Kathryn Mann

#### Abstract

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.

#### Article information

Source
Algebr. Geom. Topol., Volume 12, Number 1 (2012), 421-433.

Dates
Accepted: 18 November 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715344

Digital Object Identifier
doi:10.2140/agt.2012.12.421

Mathematical Reviews number (MathSciNet)
MR2916281

Zentralblatt MATH identifier
1268.37067

#### Citation

Mann, Kathryn. Bounded orbits and global fixed points for groups acting on the plane. Algebr. Geom. Topol. 12 (2012), no. 1, 421--433. doi:10.2140/agt.2012.12.421. https://projecteuclid.org/euclid.agt/1513715344

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