Actions of Cremona groups on CAT(0) cube complexes

Groups acting on CAT(0) cube complexes

We show that groups satisfying Kazhdan’s property (T) have no unbounded actions on nite dimensional CAT(0) cube complexes, and deduce that there is a locally CAT(−1) Riemannian manifold which is not homotopy equivalent to any nite dimensional, locally CAT(0) cube complex. AMS Classi cation numbers Primary: 20F32 Secondary: 20E42, 20G20

Partial Actions of Groups on Cell Complexes

Isometry Groups of Cat(0) Cube Complexes

Given a CAT(0) cube complex X, we show that if Aut(X) 6= Isom(X) then there exists a full subcomplex of X which decomposes as a product with R. As applications, we prove that ifX is δ-hyperbolic, cocompact and 1-ended, then Aut(X) = Isom(X) unless X is quasi-isometric to H, and extend the rank-rigidity result of Caprace–Sageev to any lattice Γ ≤ Isom(X).

Coxeter Groups Act on Cat(0) Cube Complexes

We show that any finitely generated Coxeter group acts properly discontinuously on a locally finite, finite dimensional CAT(0) cube complex. For any word hyperbolic or right angled Coxeter group we prove that the cubing is cocompact. We show how the local structure of the cubing is related to the partial order studied by Brink and Howlett in their proof of automaticity for Coxeter groups. In hi...

Actions of Picture Groups on CAT(0) Cubical Complexes

A class of groups, called picture groups, is defined. Richard Thompson’s groups F , T , and V are all picture groups. Each picture group is shown to act properly and isometrically on a CAT(0) cubical complex. In particular, all picture groups are a-T-menable. Mathematics Subject Classifications (2000). 20F65, 43A15.