[FOM] Banach Tarski Paradox/Line

Joe Shipman JoeShipman at aol.com
Mon Nov 28 00:50:22 EST 2011

The obstacle is the commutativity (or more technically the amenability which is a more general property) of the group of rigid motions of the interval (or of the real line, or of the plane, because there is no Banach-Tarski paradox in 2 dimensions either). 

The free group on two generators has a paradoxical decomposition where it is equal to a union of translates of itself and this can be used to construct a paradoxical decomposition of a space on which this group acts in a measure preserving way. Amenable groups cannot contain this group as a subgroup, but the group of rigid motions of R^3 does and the Banach-Tarski construction works.

-- JS

On Nov 27, 2011, at 4:27 PM, pax0 at seznam.cz wrote:

> Is the Banach Tarski paradox provable for the unit real interval; 
> i.e. is there a possibility for duplicating [0,1].
> If not, where is the obstacle?
> Jan Pax
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