Questions on freedom of actions

[Abstract]

**Abstract:** A topological group *G* admits a free action if
there is a
compact Hausdorff space *X* and an action of *G* on *X* such that
every non-identity *g*∈ *G* acts without fixed points. By Veech's
theorem, all locally compact groups admit a free action. On the other
side of the spectrum, there are groups that have a common fixed point
under any action; such groups are called extremely amenable. We ask
which groups other than locally compact admit free actions and how
far the property that no non-identity element acts freely under any
action is from extreme amenability. These questions may be
reformulated in terms of finite colourability of certain graphs. This
is work in progress with Vladimir Pestov.