Coarse decompositions for boundaries of cubulated CAT(0) spaces
by
Dan Guralnik
Vanderbilt University

In this talk we shall introduce a combinatorial notion of boundary ÂH for a cubing C(H) arising as the dual of a discrete w-dimensional poc-set H. ÂH has a natural median algebra structure, as well as a natural ordering whose intervals coincide with its intervals as a median algebra. Endowed with this structure, we call ÂH the Roller boundary of the cubing C(H).

When H happens to be a (discrete) G-invariant system of halfspaces in a CAT(0) space X endowed with a geometric action by a group G, we show how one can use ÂH for producing a topologically meaningful decomposition/stratification of the CAT(0) boundary ¶X of X, having much to do with both the cone and Tits topologies on ¶X.

The sets into which ¶X is decomposed are defined as the fibers of a (discontinuous) map r of ¶X into ÂH; this map is actually well-defined for any reasonably discrete halfspace system in X, but has more interesting properties in the presence of a G-action.

Finally, we provide a criterion (in terms of the image of r) for G to act co-compactly on the cubing dual to H. This criterion links the co-compactness of the action of G on C(H) with the quality of the approximation of boundary points by the halfspaces of H.

Our constructions and results provide a setting in which several issues of interest to geometric group theory are tied together: the end structure of semi-splittable groups, CAT(0) boundary topology, co-compact cubulations. In view of the results by Niblo-Reeves, Williams and Caprace, Coxeter groups supply a particularly good example of a setting to which this machinery can be applied in hopes of understanding their CAT(0) boundaries and the connections between boundary topology and, say, visual splittings of the corresponding group.

Date received: January 22, 2006