A Cubical Flat Torus Theorem and the Bounded Packing Property

We prove the bounded packing property for any abelian subgroup of a group acting properly and cocompactly on a CAT(0) cube complex. A main ingredient of the proof is a cubical flat torus theorem. This ingredient is also used to show that central HNN extensions of maximal free-abelian subgroups of compact special groups are virtually special, and to produce various examples of groups that are not cocompactly cubulated.Comment: 14 pages, 2 figures, submitted May 2015 Minor corrections and swapped sections 2 and 3 Corrected an unfortunate typo in Theorem 2.1 - the hypothesis that the cube complex be finite dimensional has now been adde

Quasiconvexity and relatively hyperbolic groups that split

We explore the combination theorem for a group G splitting as a graph of relatively hyperbolic groups. Using the fine graph approach to relative hyperbolicity, we find short proofs of the relative hyperbolicity of G under certain conditions. We then provide a criterion for the relative quasiconvexity of a subgroup H depending on the relative quasiconvexity of the intersection of H with the vertex groups of G. We give an application towards local relative quasiconvexity