Fixed point theorems and rigidity

Martin Bridson, Oxford University

I shall begin by sketching the proof that SL(n,Z) and the
corresponding index-2 subgroup of Aut(Fn) admit no non-trivial action
by homeomorphisms on any sphere of dimension less than n-1 (the
dimension of the standard linear action). This is joint work with
Karen Vogtmann. I shall explain how some of the ideas that go into
this proof can be adapted to prove that there exist finitely presented
groups that cannot act non-trivially on any finite-dimensional mod-p
acyclic manifold. Finally, I shall explain joint work with
Arzhantseva, Januszkiewicz, Leary, Minasyan and Swiatkowski in which
we settle a question of Kropholler by constructing finitely generated
groups that cannot act without a global fixed point on any
contractible, finite dimensional CW-complex.