Ph.D Thesis | |

Ph.D Student | Guralnik Dan |
---|---|

Subject | Coarse Decompositions of Boundaries for CAT(0) Groups |

Department | Department 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 H^{0}, where H is
the halfspace system dual to C. The ‘boundary’ we propose is the set of
almost-equality classes of H^{0}. 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 H^{0}) 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.