FOM: My BT post
David Ross
ross at math.hawaii.edu
Thu Dec 11 13:06:38 EST 1997
Good grief, I certainly never expected my remark to be held up as
a paradigm of a bad post!
I think Balwin and Steel made my meaning clear (as I apparently hadn't),
but let me add that I'd hoped readers of this list would be familiar
with the result (due to Stan Wagon, early 80s) that Day's conjecture holds
for rigid motions in R^n, n>2. In other words, a subgroup of such motions
can be used to paradoxically decompose a sphere if and only if it
contains F2 (iff the subgroup is not amenable, etc - Wagon listed quite
a few equivalents, as I recall). While the one volume paradox - most
troubling to barbers - might go away if we restricted our attention to
Borel sets, most of the others would not. In fact, as the passage (in
the proof of BT) from existence of F2 in this group to the Hausdorff
paradox is almost identical to the usual proof for existence of
a nonmeasurable subset of [0,1], I would suggest that that the BT
paradox is no more problematic from the standpoint of which sets to
accept as is the simpler nonmeasurability argument.
BTW, I never meant this as an indictment of the interesting programme of
restricting our atention to Borel sets, merely as an indictment of the
choice of one of the motivating examples. (I also never meant this as
an indictment of the free group on two generators, which is one of my
favorite mathematical objects.)
- David (ross at tarski.math.hawaii.edu)
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