Connectedness properties and splittings of groups with isolated flats

Abstract

In this paper we study CAT(0) groups and their splittings as graphs of groups. For one-ended CAT(0) groups with isolated flats we prove a theorem characterizing exactly when the visual boundary is locally connected. This characterization depends on whether the group has a certain type of splitting over a virtually abelian subgroup. In the locally connected case, we describe the boundary as a tree of metric spaces in the sense of Świątkowski. A significant tool used in the proofs of the above results is a general convex splitting theorem for arbitrary CAT(0) groups. If a CAT(0) group splits as a graph of groups with convex edge groups, then the vertex groups are also CAT(0) groups.

37 pages. The introduction has been substantially rewritten. A claimed result from an earlier version about the semistability of all CAT(0) groups with isolated flats has been removed because the proof contained an error. A correct proof of that result can be found in the article arXiv:1904.12947. Some theorem numbers may have changed

References in corpus (2)

• Relatively Hyperbolic Groups have Semistabile Fundamental Group at Infinity
• Boundary classification and 2-ended splittings of groups with isolated flats

Cited by in corpus (3)

• Local cut points and splittings of relatively hyperbolic groups
• Cannon--Thurston maps for CAT(0) groups with isolated flats
• Equivariant blowups of bounded parabolic points