Collapses and Z-Compactifications of CAT(0) Cube Complexes

Mr. Daniel Gulbrandsen
University of Wisconsin-Milwaukee
Dissertator

We extend the notion of collapsibility to non-compact cube complexes and prove collapsibility of locally-finite CAT(0) cube complexes. Namely, we construct such a cube complex X out of nested convex compact subcomplexes {C_i} with the property that C_i collapses to C_{i-1} for all i>0.

We then define retractions r_i between the compacta C_i and use these as bonding maps to construct an inverse sequence yielding the inverse limit space Y. We show that Y is a Z-compactification of X. The process produces a new Z-boundary, called the cubical boundary. We will finish with some examples and open questions.

Advisor: Prof. Craig Guilbault

Committee Members:
Profs. Boris Okun, Christopher Hruska, Jonah Gaster, and Jeb Willenbring