# The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes

Abstract : We define a bounded cohomology class, called the {\em median class}, in the second bounded cohomology -- with appropriate coefficients -- of the automorphism group of a finite dimensional CAT(0) cube complex X. The median class of X behaves naturally with respect to taking products and appropriate subcomplexes and defines in turn the {\em median class of an action} by automorphisms of X. We show that the median class of a non-elementary action by automorphisms does not vanish and we show to which extent it does vanish if the action is elementary. We obtain as a corollary a superrigidity result and show for example that any irreducible lattice in the product of at least two locally compact groups with finitely many connected components acts on a finite dimensional CAT(0) cube complex X with a finite orbit in the Roller compactification of X. In the case of a product of Lie groups, the Appendix by Caprace allows us to deduce that the fixed point is in fact inside the complex X.
Type de document :
Pré-publication, Document de travail
2015

https://hal.archives-ouvertes.fr/hal-01145335
Contributeur : Indira Chatterji <>
Soumis le : jeudi 23 avril 2015 - 20:19:06
Dernière modification le : jeudi 3 mai 2018 - 13:32:58

### Identifiants

• HAL Id : hal-01145335, version 1
• ARXIV : 1212.1585

### Citation

Indira Chatterji, Talia Fernos, Alessandra Iozzi. The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes. 2015. 〈hal-01145335〉

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