In this paper, \(X\) is a CAT(0) space and \(\Gamma\) is a group acting isometrically, properly discontinuously and cocompactly on \(X\). For \(k\geq 2\), a \(k\)-flat in \(X\) is an isometrically embedded Euclidean space \(\mathbb{E}^n\) in \(X\). \(\text{Flat}(X)\) denotes the space of all flats in \(X\) with the topology of Hausdorff convergence on bounded sets. The space \(X\) is said to ‘have isolated flats’ with respect to the \(\Gamma\) action if \(X\) contains an equivariant collection \(\mathcal F\) of flats such that \(\mathcal F\) is closed and isolated in \(\text{Flat}(X)\), and each flat in \(X\) is contained in a uniformly bounded tubular neighborhood of some flat in the collection \(\mathcal F\). CAT(0) spaces with isolated flats were first considered by Kapovich-Leeb and by Wise, independently.
One of the main objects of the paper under review is the exploration of various notions of isolated flats and the proof that these notions are equivalent.
The authors obtain the following results: Theorem 1: Let \(X\) be a CAT(0) space and \(\Gamma\) a group acting isometrically, properly discontinuously and cocompactly on \(X\). Then, the following are equivalent: (1) \(X\) has isolated flats; (2) the Tits boundary of \(X\) is a disjoint union of isolated points and standard Euclidean spheres; (3) \(X\) is a relatively hyperbolic space with respect to a family of flats; (4) \(\Gamma\) is a relatively hyperbolic group with respect to a collection of virtually Abelian subgroups of rank \(\geq 2\).
The authors also obtain the following, as a consequence of Theorem 1: Theorem 2: Let \(X\) be a CAT(0) space and \(\Gamma\) a group acting isometrically, properly discontinuously and cocompactly on \(X\) with isolated flats. Then, the following properties hold: (1) quasi-isometries of \(X\) map maximal flats to maximal flats; (2) a finitely generated subgroup \(H\leq\Gamma\) is undistorted if and only if it is quasiconvex (with respect to the CAT(0) action); (3) the geometric (i.e. the visual) boundary of \(X\) is a group invariant of \(\Gamma\); (4) \(\Gamma\) satisfies the strong Tits alternative: every subgroup of \(\Gamma\) either is virtually Abelian or contains a free subgroup of rank two; (5) \(\Gamma\) is biautomatic.
In an appendix written jointly with M. Hindawi, the authors show that many of the results of this article extend to a more general class of CAT(0) spaces, where the collection \(\mathcal F\) of flat subspaces is replaced by a collection of closed convex subspaces of \(X\).