Tree-graded space

A geodesic metric space [math]\displaystyle{ X }[/math] is called a tree-graded space with respect to a collection of connected proper subsets called pieces, if any two distinct pieces intersect in at most one point, and every non-trivial simple geodesic triangle of [math]\displaystyle{ X }[/math] is contained in one of the pieces. If the pieces have 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 {{harvs|txt|last1 = Druţu | first1 = Cornelia | author1-link = Cornelia Druţu

References

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