Homotopy of ends and boundaries of CAT(0) groups. (Q2503125)

The authors show that the group \(H\) given by the presentation \(\langle a,b,c,d \mid [a,b]=[b,c]=[c,d]=1\rangle\) can be given a CAT(0) geometry, is 1-ended, is simply-connected at infinity but has a non-path connected boundary with non-trivial fundamental group. The groups \(H\times\mathbb{Z}^n\) for \(n\geq 1\) are shown to be \(n\)-connected at infinity but have boundary with trivial homotopy in dimensions 0 through \(n-1\) and nontrivial \(n\)-th homotopy group. In particular it is shown that all parabolic semidirect products of the free group of rank 2 and \(\mathbb{Z}\) have a non-path connected boundary. Then the authors construct another example of a group \(G\) which is simply connected at infinity and has non-simply connected boundary. This group acts geometrically on a 3-dimensional CAT(0) space with 2-dimensional boundary.