Actions and homomorphisms of topological full groups

To any group or pseudogroup of homeomorphisms of the Cantor set one
can associate a larger (countable)  group, called the topological
full group. It is a complete invariant of the groupoid of germs of
the underlying action (every isomorphism between full groups is
implemented by a conjugacy of the pseudogroup).
First I'll state a result relating  the growth of the orbits of a
pseudogroup to a combinatorial fixed point property of its full
group, and explain an application related to co-amenability and
growth of Schreier graphs of finitely generated groups. Next I will
discuss a theorem on the possible actions on topological full groups
on compact spaces, and apply it to show that not only isomorphisms
but arbitrary homomorphisms between full groups are often implemented
at the level of the groupoids. These results are proven working in the
Chabauty space.