Tree-graded space

A geodesic metric space $X$ is called tree-graded space, with respect to a collection of connected proper subsets called pieces, if any two distinct pieces intersect by at most one point, and every non-trivial simple geodesic triangle of $X$ is contained in one of the pieces.

Thus, for pieces of bounded diameter, tree-graded spaces behave like real trees in their coarse geometry (in the sense of Gromov) while allowing non-tree-like behavior within the pieces.

Tree-graded spaces were introduced by Druţu & Sapir (2005) in their study of the asymptotic cones of hyperbolic groups.

References

• Druţu, Cornelia; Sapir, Mark (2005), "Tree-graded spaces and asymptotic cones of groups", Topology 44 (5): 959–1058, doi:10.1016/j.top.2005.03.003, MR 2153979.