2017/11/30 by Elia Fioravanti
Mathematics · #Advanced Topology and Set Theory #Bounded function #Cohomology #Combinatorics #Compactification (mathematics) #Corollary #Cube (algebra) #Discrete mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Lattice (music) #Mathematical analysis #Mathematics #Pure mathematics #Rank (graph theory) #math.GR #math.GT #math.MG
paper · pdf · doi:10.1016/j.aim.2019.06.019
published as Adv. Math. 352 (2019), 1206-1252 · 46 pages, 3 figures; final version, to appear on Adv Math
arxiv created 2019/06/17 · openalex publication_date 2019/06/28 · arxiv updated 2019/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Finite rank median spaces are a simultaneous generalisation of finite dimensional \rm CAT(0) cube complexes and real trees. If Γ is an irreducible lattice in a product of rank one simple Lie groups, we show that every action of Γ 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 \rm 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 HFD.