Groups acting on \(\text{CAT}(0)\) cube complexes. (English) Zbl 0887.20016
Summary: We show that groups satisfying Kazhdan’s property (T) have no unbounded actions on finite dimensional \(\text{CAT}(0)\) cube complexes, and deduce that there is a locally \(\text{CAT}(-1)\) Riemannian manifold which is not homotopy equivalent to any finite dimensional, locally \(\text{CAT}(0)\) cube complex.
MSC:
| 20F65 | Geometric group theory |
| 57M07 | Topological methods in group theory |
| 57M05 | Fundamental group, presentations, free differential calculus |
| 57N10 | Topology of general \(3\)-manifolds (MSC2010) |
| 20E08 | Groups acting on trees |
Keywords:
Kazhdan’s property (T); Tits buildings; hyperbolic geometries; Riemannian manifolds; group actions on cube complexesReferences:
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