# Tree-graded space

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A geodesic metric space 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 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 Template:Harvtxt in their study of the asymptotic cones of hyperbolic groups.

## References

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