For each d, we construct CAT(0) cube complexes on which Cremona groups of rank d act by isometries. From these actions deduce new and old group-theoretical dynamical results about groups. In particular, study the behavior irreducible components exceptional loci. This leads to proofs regularization theorems, such as with property FW. We also find constraints degree growth for non-pseudo-regulari...

All complete, axially symmetric surfaces of constant mean curvature in R lie in the one-parameter family Dτ of Delaunay surfaces. The elements of this family which are embedded are called unduloids; all other elements, which correspond to parameter value τ ∈ R, are immersed and are called nodoids. The unduloids are stable in the sense that the only global constant mean curvature deformations of...

There is a well-known theory of differentiable cohomology H diff (G, V ) of a Lie group G with coefficients in a topological vector space V on which G acts differentiably. This is developed by Blanc in [Bl]. It is very desirable to have a theory of differentiable cohomology for a (possibly infinite-dimensional) Lie group G, with coefficients in an arbitrary abelian Lie group A, such that the gr...

We specify exactly which groups can act geometrically on CAT(0) spaces whose visual boundary is homeomorphic to either a circle or a suspension of a Cantor set.

We construct CAT(0) groups containing subgroups whose Dehn functions are given by x , for a dense set of numbers s ∈ [2,∞). This significantly expands the known geometric behavior of subgroups of CAT(0) groups.

In this paper we establish a rigid connection between two classical objects: the R.Thompson group (the group of all piece-wise linear homeomorphisms of the unit interval with finitely many dyadic break points and all slopes powers of 2) and the Dunce hat (the topological space obtained from the triangle ABC by gluing AB, BC and AC). We prove that a diagram group of a directed 2-complex contains...