Ph.D Thesis

Ph.D StudentGuralnik Dan
SubjectCoarse Decompositions of Boundaries for CAT(0) Groups
DepartmentDepartment of Mathematics
Supervisors PROFESSOR EMERITUS Bronislaw Wajnryb
PROF. Michah Sageev


In this thesis we construct a combinatorial notion of boundary for w-dimensional cubings (non-positively curved cube complexes) using the presentation of a cubing as the principal component of its double-dual H0, where H is the halfspace system dual to C. The ‘boundary’ we propose is the set of almost-equality classes of H0. It suffices to consider w-dimensional discrete poc-sets H, as those are dual to w-dimensional cubings via the well-known Sageev-Roller duality.

We prove that R(H) is organized in a hierarchical manner corresponding to the interactions between the Tychonoff closures (in H0) of its elements, and construct notions of g.c.d. and co-dimension for elements of R(H).

Next, we apply this tool to the study of a discrete invariant w-dimensional halfspace system in a CAT(0) space with a geometric action by a group G. We show how to construct a map r from the boundary X of X into R(H), producing a stratification of X induced from the hierarchy of R(H). The map r is shown to provide connections between connectivity properties of R(H) (as a graph) and connectivity properties of X as a path metric space (when taken with the Tits metric). The main result of this work is a characterization, in terms of the image of r, of the situation when G acts co-compactly on the cubing dual to the halfspace system H. We hope the tools constructed herein will prove effective in showing relations between connectivity properties of boundaries of CAT(0) groups and abelian splittings of such groups.