### 18 december 2006,
Tsachik Gelander (Hebrew University, Israel),
On a conjecture by Margulis and Soifer

**Abstract.**
For n>1, a generic n--tuple of elements in a connected
compact non-abelian
Lie group G generates a free group. Margulis and Soifer conjectured
that
every such tuple can be slightly deformed to one which generates a
group
which is not virtually free. I will explain a proof of this
conjecture, and
actually show that for n>2 and for an arbitrary dense subgroup D,
with some restriction
on the minimal size of a generating set, the set of deformations of
F_n whose image is D is dense in
the variety of all deformations. The proof relies on the product
replacement
method. Using the same ideas I will also give a proof of a conjecture
of
Goldman on the ergodicity of the action of Out(F_n) on Hom(F_n,G)/G
when
n>3. For n=2, I will explain how to produce for any pair (a,b) an
arbitrarily close pair (a',b') which generates an infinite group
which has
Serre property FA and in particular is not virtually free.