The median class and superrigidity of actions on CAT(0) cube complexes. With an appendix by Pierre-Emmanuel Caprace
(2016) Journal of Topology and Analysis — Vol. 9, p. 349-400 (2016)
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- Abstract
- We define a bounded cohomology class, called the 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 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 what 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 connected groups 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 . In the course of the proof, we construct a Γ -equivariant measurable map from a Poisson boundary of Γ with values in the non-terminating ultrafilters on the Roller boundary of X .
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Caprace, P.-E., Chatterji, I., Fernos, T., & Iozzi, A. (2016). The median class and superrigidity of actions on CAT(0) cube complexes. With an appendix by Pierre-Emmanuel Caprace. Journal of Topology and Analysis, 9, 349-400. https://doi.org/10.1112/jtopol/jtu025 (Original work published 2016)