Groups, Geometry, and Dynamics

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Volume 8, Issue 2, 2014, pp. 467–478
DOI: 10.4171/GGD/234

Published online: 2014-07-08

Amenable groups with a locally invariant order are locally indicable

Peter Linnell[1] and Dave Witte Morris[2]

(1) Virginia Tech, Blacksburg, USA
(2) University of Lethbridge, Canada

We show that every amenable group with a locally invariant partial order has a left-invariant total order (and is therefore locally indicable). We also show that if a group $G$ admits a left-invariant total order, and $H$ is a locally nilpotent subgroup of $G$, then a left-invariant total order on $G$ can be chosen so that its restriction to $H$ is both left-invariant and right-invariant. Both results follow from recurrence properties of the action of $G$ on its binary relations.

Keywords: Locally invariant order, left-invariant order, left-orderable group, right-orderable group, recurrent order, locally indicable, amenable group

Linnell Peter, Witte Morris Dave: Amenable groups with a locally invariant order are locally indicable. Groups Geom. Dyn. 8 (2014), 467-478. doi: 10.4171/GGD/234