They are used in computer science, in the reconstruction of phylogenetic trees, but not so well known by mathematicians.
Today, I give an expository talk with definitions but no proofs. Proofs will come later on.
1. Injective metric spaces
Definition 1 A metric space is injective if every for every metric space and every , every -Lipschitz map has a -Lipschitz extension .
Example 1 The real line is injective.
An observation of Mc Shane and Whitney in 1934. Put .
Example 2 is injective for every set .
Example 3 Metric trees are injective.
1.2. Elementary properties
If is injective, then
- is complete.
- is geodesic.
- is contractible. In fact, has a geodesic bicombing, i.e. a family of geodesics such that
(use the Kuratowski embedding, and the distance decreasing retraction of onto ).
- Every triple spans a geodesic tripod.
- admits isoperimetric inequalities of Euclidean type for integral cycles in any dimension (Gromov, Wenger).
Property 3. reminds us of nonpositive curvature. Property 4. sound like strong negative curvature, but with strong non uniqueness of geodesics.
1.3. Elementary characterizations
Proposition 2 (Aronszajn, Panitchpakdi) The following are equivalent:
- is injective
- is an absolute -Lipschitz retract.
- is hyperconvex, i.e. for every collection such that all , the intersection is non empty.
It follows that intersections of balls in injective metrics spaces are injective.
2. Injective hulls
J. R. Isbell 1964, rediscovered by A. Dress in 1984, under the name of tight spans (aka injective injective envelopes,…), [I].
Theorem 3 (Isbell, Dress) For every metric space , there exists an injective metric space and an isometric embedding with the following universal property: every isometric embedding of in a injective metric space factors via . Let us call it the injective hull of .
Proof: The proof provides an explicit construction. Let be the space of functions such that . Distances to points belong to . Let be the set of minimal elements of the partially ordered set (akaextremal functions). Every is -Lipschitz. If , , then is bounded, so defines a metric on . Distances to points belong to and this gives the isometric embedding of in . Note that so one can think of extremal functions are distance functions to themselves, i.e. points which do not belong to .
Example 4 If has two points, is an interval.
Example 5 If has three points, is a tripod.
Example 6 If has four points, is a rectangle sitting diagonally in with antennas sticking out.
Example 7 If is finite, is a polyhedron made of finitely many cells isometric to polytopes of , of various dimensions .
Example 8 If is the -cycle, .
2.2. Injective hulls and phylogenetic trees
Start be a finite simplicial tree. One can reconstruct the tree from the metric induced on terminal vertices. Indeed, the tree is the injective hull of its leaves. Assume a metric is given on a finite set. This metric is expected to arise from a tree, with some noise. Compute the -skeleton of the injective hull, this is a good approximation of a tree (Andreas Dress and collaborators).
3. Hyperbolic spaces and groups
Definition 4 (Gromov) Let be a geodesic metric space. is -hyperbolic if every triangle has a map to a tripod such that inverse images of points have diameters .
Theorem 5 (Lang 2004) If is -hyperbolic, then is a -neighborhood of . In particular, is hyperbolic.
By contrast, the injective hull of Euclidean plane is .
Let be a finitely generated group. Fix a word metric . Then acts isometrically on . The actions is proper in the following weak sense: for every bounded set , the set of such that is finite. It is co-bounded.
Theorem 6 (Lang, Moezzi 2010) Let be a hyperbolic group. Then is a locally finite polyhedral complex with finitely many isometry types of celles, and acts properly and cocompactly on by cellular isometries.
In other words, there is a nice model of . It has nonpositive curvature features, but it is not CAT. If it is -dimensional, it can be remetrized to be CAT.
Proof: For , define
Cannon discovered that hyperbolic groups have finitely many cones. This gives the upper bound . I believe that the injective hull has dimension smaller that Rips’ complex (by a factor of ).
4. Fixed point theorems
Theorem 7 (Lang) Let be an injective metric space. Let be a group of isometries of .
- If has bounded orbits, then is non empty.
- If is non empty, is injective and hence contractible.
Corollary 8 Let be the first barycentric subdivision of . If is a hyperbolic group, then is a model for the classifying space for proper actions of .
Next time, I will give more details and tell more about injective hulls of discrete metric spaces.
Krauthgamer: how does injective hull behave with respect to quasiisometries ? Answer: It is -Lipschitz with respect to Gromov-Hausdorff distance.
Krauthgamer: what about the endpoint set of a tree ?