Title: Superrigidity of actions on finite rank median spaces,
Abstract: Finite rank median spaces are a simultaneous generalisation of finite
dimensional CAT(0) cube complexes and real trees. If $\Gamma$ is an irreducible
lattice in a product of rank one simple Lie groups, we show that every action
of $\Gamma$ on a complete, finite rank median space has a global fixed point.
This is in sharp contrast with the behaviour of actions on infinite rank median
spaces.
The fixed point property is obtained as corollary to a superrigidity result;
the latter holds for irreducible lattices in arbitrary products of compactly
generated groups.
In previous work, we introduced "Roller compactifications" of median spaces;
these generalise a well-known construction in the case of cube complexes. We
provide a reduced $1$-cohomology class that detects group actions with a finite
orbit in the Roller compactification. Even for CAT(0) cube complexes, only
second bounded cohomology classes were known with this property, due to
Chatterji-Fernós-Iozzi. As a corollary, we observe that, in Gromov's density
model, random groups at low density do not have Shalom's property $H_{FD}$.