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

Abstract

We extend the notion of collapsibility to non-compact 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 $\{C_i\}_{i=0}^\infty$ with the properties that $X=\cup_{i=0}^\infty C_i$ and $C_i$ collapses to $C_{i-1}$ for all $i\ge 1$. We then define bonding maps $r_i$ between the compacta $C_i$ and construct an inverse sequence yielding the inverse limit space $\varprojlim\{C_i,r_i\}$. This will provide a new way of Z-compactifying $X$. In particular, the process will yield a new Z-boundary, called the cubical boundary.