This lecture start a Metric Geometry Day held in Orsay on july 8th, 2013.
Speakers : U. Lang (Zurich), P. Haissinsky (Toulouse), V. Chepoi (Marseille), V. Shchur (Orsay, PhD defense).
Weak notions of global nonpositive curvature
1. Geodesic bicombings
Definition 1 A geodesic bicombing on a geodesic metric space is a map
such that is a unit speed geodesic for all .
Say the bicombing is convex if is convex for all .
Say the bicombing is consistent if the image of is contained in the image of if and belong to the image of .
Say the bicoming is conical if
Remark 1 1. Although it looks similar, the conicality condition does not follow from convexity. In fact, conical + consistent convex.
2. If is a convex subset of a Banach space, the obvious linear bicombing is convex and consistent.
3. If has a conical geodesic bicombing, and is a 1-Lipschitz retraction, the composed map is again a conical geodesic bicombing on .
Example 1 In with the -norm, consider a triangle , and for the union of two sides an oblique square contained in . The vertical projection is 1-Lipschitz
1.2. A hierarchy of global nonpositive curvature conditions
1. (triangle comparison with Euclidean plane) is the strongest.
2. Busemann : the distance function is convex along pairs of geodesics.
3. has a convex and consistent geodesic bicombings.
4. has a convex geodesic bicombing.
5. has a conical geodesic bicombing.
For instance, Banach spaces satisfy 3. A Banach space satisfies 2 iff the norm is strictly convex (making geodesics unique). A Banach space is iff it is a Hilbert space.
3 and 1 are stable under limits but 1 is not.
In the sequel, I will give two results relating 5 to 4 and 4 to 3. Furthermore, I will prove that injective metric spaces satisfy 5.
1.3. Injective metric spaces
Definition 2 A metric space is injective if for every pair of metric spaces, every 1-Lipschitz map has a 1-Lipschitz extension .
This notion goes back to the 1930’s.
Example 2 , , metric trees are injective.
Isbell 1964 : Every metric space has a unique injective hull. This shows that there are many examples.
2. Improving conical to convex
2.1. The result
Theorem 3 If is proper and has a conical geodesic bicombing, then has a convex geodesic bicombing.
Call a geodesic bicombing -discretely convex if
holds for all which belong to .
By definition, conical -discretely convex. This can be improved into a -discretely convex geodesic bicombing as follows. Given , join them, join to , join to , join to , and so on. This produces Cauchy sequences, converging to a new bicombing, which is -discretely convex.
By compactness, on produces a limit bicombing, which is convex.
3. Improving convex to consistent
3.1. The result
Theorem 4 If has finite combinatorial dimension and has a convex geodesic bicombing , then is consistent and unique.
3.2. Isbell’s injective hull
Definition 5 Let be a metric space. Let
Elements of , called by Isbell extremal functions, look like distance functions to set, from far away. The sup norm defined a metric on , in which is isometrically embedded. is injective, it is the injective hull of .
For finite sets , is a polyhedron, is a subcomplex of the boundary of . For instance, for a 2-point space, is an interval. For a 3-point space, is a tripod. For a 4-point space, is a rectangle modelled on with 4 antennas attached. For 5-point spaces, several combinatorial types occur, again consisting of -rectangles and antennas. Some are planar, others are not.
3.3. Combinatorial dimension
Definition 6 (Dress, 1984) Let be a metric space. The combinatorial dimension of is the supremum of the dimensions of , a finite subset of .
Example 3 A Banach space has finite combinatorial dimension if and only if the unit ball is a polyhedron.
Metric trees have combinatorial dimension 1.
I will not prove the following Lemma (it is a bit technical).
Lemma 7 If has finite combinatorial dimension, then for all , there exists such that
for all , .
This clearly fails for Euclidean plane, but holds for the line.
3.4. Proof of Theorem 2
Consider , . This function is convex along axes. By the Lemma, in the small, we get convexity along the diagonal. This implies uniqueness of geodesics, and therefore consistency.
Theorem 8 Let be a hyperbolic group, equipped with some word metric. Then is a proper finite dimensional polyhedral complex with finitely many isometry types of cells. acts properly and cocompactly on .
It follows that has a unique -invariant convex and consistent geodesic bicombing. This is weaker that , but not that bad.
What if one changes the metric on cells to a Euclidean metric, does it improve ? In dimension 2, this works well and produces a space. In higher dimensions, it may work. In progress.