The median class and superrigidity of actions on CAT(0) cube complexes

By Indira Chatterji, Talia Fern'os, A. Iozzi. 2012

83 citations recorded by the source corpus.

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 .

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e5b9a37d3b2a2fd7cf38b348d3c17633625aace2

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In Geometry and Topology

1. Anthony Genevois
2. Alexandre Martin
3. Hung Cong Tran

In Mathematical Physics

1. Anthony Genevois
2. Alexandre Martin
3. Hung Cong Tran

Citations in the corpus

• Automorphisms of graph products of groups from a geometric perspective (2018, background): “Following [10], we say that the action is non-elementary if G does not have a finite orbit in Y ∪ ∂∞Y .”