I will speak about a joint work with A. Karlsson and G.A. Margulis.
Since Margulis proved his remarkable supper-rigidity theorem, various
extensions and generalizations of it were proved by various people.
The supper-rigidity theorem can be read as follows. Let G be a locally
compact group and L a lattice in G. Let X be a space on which L acts by
isometries. Then, under an appropriate assumptions on G,L,X and the action,
the action extends to G. In Margulis’ classical theorem G is a higher rank
semisimple Lie group, L an irreducible lattice, X a symmetric space of
non-compact type and the action is unbounded and Zariski dense (in
Isom(X)).
We proved a supper-rigidity theorem for irreducible lattices L in a
product of general locally compact groups G=G_1x…xG_n (n>1), where X is
a general metric space whose metric satisfies some convexity assumptions.
This generalize a recent theorem of N. Monod who establish the same result
when X is a CAT(0) space.
Our proof uses the notion of generalized harmonic maps, and is influenced
by a preprint of Margulis (from ~ 1990) about supper-rigidity for
commensurable groups.
This generalize a recent theorem of N. Monod who establish the same result when X is a CAT(0) space.
Our proof uses the notion of generalized harmonic maps, and is influenced by a preprint of Margulis (from ~ 1990) about supper-rigidity for commensurable groups.