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Separation with restricted families of sets
Given a finite -element set , a family of subsets is said to separate if any two elements of are separated by at
least one member of . It is shown that if ,
then one can select members of that
separate . If for some , then
members of
are always sufficient to separate all pairs of elements of that are
separated by some member of . This result is generalized to
simultaneous separation in several sets. Analogous questions on separation by
families of bounded Vapnik-Chervonenkis dimension and separation of point sets
in by convex sets are also considered.Comment: 13 page
Fixed points for actions of Aut(Fn) on CAT(0) spaces
For n greater or equal 4 we discuss questions concerning global fixed points
for isometric actions of Aut(Fn), the automorphism group of a free group of
rank n, on complete CAT(0) spaces. We prove that whenever Aut(Fn) acts by
isometries on complete d-dimensional CAT(0) space with d is less than 2 times
the integer function of n over 4 and minus 1, then it must fix a point. This
property has implications for irreducible representations of Aut(Fn), which are
also presented here. For SAut(Fn), the unique subgroup of index two in Aut(Fn),
we obtain similar results
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