G. Christopher Hruska
G. Christopher Hruska
Ph.D. (2002) Cornell University
Abstract: The mildest way that a nonpositively curved space can fail to be negatively curved is for it to contain only a sparse collection of isolated flat Euclidean subspaces. The concept of a CAT(0) space whose flat planes are isolated is implicit in work of Michael Kapovich and Bernhard Leeb and of Daniel Wise and has also been studied by Bruce Kleiner. In this dissertation we introduce the Isolated Flats Property, which makes this notion explicit, and we show that several important results about Mikhail Gromov's $\delta$-hyperbolic spaces extend to the class of CAT(0) spaces with this property.
More specifically, we consider a large class of groups which act properly and cocompactly by isometries on CAT(0) spaces with the Isolated Flats Property. We show that the family of groups in question includes all those
groups which act on 2-dimensional complexes with the Isolated Flats Property as well as all geometrically finite subgroups of $\Isom ( \Hyp^n )$. We also show that for each such group there is a well-defined notion of a boundary at infinity and an intrinsic notion of a subgroup being quasiconvex. These results were established by Gromov in the negatively curved setting and do not extend to general nonpositively curved spaces.
It is reasonable to interpret the present results as indicating that groups which act properly and cocompactly by isometries on CAT(0) spaces with isolated flats are very nearly word hyperbolic. In fact, much of the inspiration for the present theory comes from a philosophy that spaces with isolated flats are hyperbolic relative to their flat Euclidean subspaces.
Our main theme is to formulate relative versions of several of the basic properties of hyperbolic spaces. For instance in the presence of the Isolated Flats Property, one can often conclude that geodesic triangles are thin relative to flats and that pairs of quasigeodesics fellow travel relative to flats in a suitable sense. In the setting of CAT(0) 2-complexes, we show that each of these relative properties is equivalent to the Isolated Flats Property.