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).

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

In Bartels–Lück [1] we introduced the concept of transfer reducible groups with respect to a family of subgroups. This definition is somewhat technical and recalled as Definition 0.4 below. We showed that groups that are transfer reducible over the family of virtually cyclic subgroups satisfy the Farrell–Jones Conjecture with coefficients in an additive category. For further explanations about ...

We show that if G is a stable group of finite weight with no infinite, infinite-index, chains of definable subgroups, then G is superstable of finite U rank. Combined with recent work of Palaćın and Sklinos, we conclude that (Z,+, 0) has no proper stable expansions of finite weight. A corollary of this result is that if P ⊆ Zn is definable in a finite dp-rank expansion of (Z,+, 0), and (Z,+, 0,...

We study groups acting on CAT (0) square complexes. In particular we show if Y is a nonpositively curved (in the sense of A. D. Alexandrov) finite square complex and the vertex links of Y contain no simple loop consisting of five edges, then any subgroup of π1(Y ) either is virtually free abelian or contains a free group of rank two. In addition we discuss when a group generated by two hyperbol...

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