Notes of Urs Lang’s july 2013 lecture

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

1.1. Definition

Definition 1 A geodesic bicombing on a geodesic metric space {X} is a map

\displaystyle  \begin{array}{rcl}  \sigma : X\times X\times [0,1]\rightarrow X \end{array}

such that {\sigma_{xy}} is a unit speed geodesic for all {x,y}.

Say the bicombing is convex if {s\mapsto d(\sigma_{xy}(s),\sigma_{x'y'}(s))} is convex for all {x,y,x',y'}.

Say the bicombing is consistent if the image of {\sigma_{x'y'}} is contained in the image of {\sigma_{xy}} if {x'} and {y'} belong to the image of {\sigma_{xy}}.

Say the bicoming is conical if

\displaystyle  \begin{array}{rcl}  d(\sigma_{xy}(s),\sigma_{x'y'}(s))\leq (1-s)d(x,x')+sd(y,y'). \end{array}

Remark 1 1. Although it looks similar, the conicality condition does not follow from convexity. In fact, conical + consistent {\Rightarrow} convex.

2. If {X} is a convex subset of a Banach space, the obvious linear bicombing is convex and consistent.

3. If {\bar{X}} has a conical geodesic bicombing, and {\rho:\bar{X}\rightarrow X} is a 1-Lipschitz retraction, the composed map is again a conical geodesic bicombing on {X}.

Example 1 In {{\mathbb R}^2} with the {\ell_{\infty}}-norm, consider a triangle {\bar{X}}, and for {X} 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. {CAT(0)} (triangle comparison with Euclidean plane) is the strongest.

2. Busemann : the distance function is convex along pairs of geodesics.

3. {X} has a convex and consistent geodesic bicombings.

4. {X} has a convex geodesic bicombing.

5. {X} 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 {CAT(0)} 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 {X} is injective if for every pair {A\subset B} of metric spaces, every 1-Lipschitz map {A\rightarrow X} has a 1-Lipschitz extension {B\rightarrow X}.

This notion goes back to the 1930’s.

Example 2 {{\mathbb R}}, {\ell_{\infty}}, 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 {X} is proper and has a conical geodesic bicombing, then {X} has a convex geodesic bicombing.

2.2. Proof

Call a geodesic bicombing {\sigma} {\frac{1}{n}}-discretely convex if

\displaystyle  \begin{array}{rcl}  d(\sigma_{xy}(s),\sigma_{x'y'}(s))\leq (1-\lambda)d(\sigma_{xy}(r),\sigma_{x'y'}(r))+\lambda(\sigma_{xy}(t),\sigma_{x'y'}(t)),\quad s=(1-\lambda)r+\lambda t. \end{array}

holds for all {r<s<t} which belong to {\frac{1}{n}{\mathbb Z}\cap[0,1]}.

By definition, conical {\Rightarrow} {\frac{1}{2}}-discretely convex. This can be improved into a {\frac{1}{3}}-discretely convex geodesic bicombing as follows. Given {x,y}, join them, join {x} to {y_1=\sigma_{xy}(\frac{2}{3})}, join {\sigma_{xy_1}(\frac{1}{2})} to {y}, join {x} to {y_2=sigma_{y_1y}(\frac{1}{2})}, and so on. This produces Cauchy sequences, converging to a new bicombing, which is {\frac{1}{3}}-discretely convex.

By compactness, on produces a limit bicombing, which is convex.

3. Improving convex to consistent

3.1. The result

Theorem 4 If {X} has finite combinatorial dimension and has a convex geodesic bicombing {\sigma}, then {\sigma} is consistent and unique.

3.2. Isbell’s injective hull

Definition 5 Let {X} be a metric space. Let

\displaystyle  \begin{array}{rcl}  \Delta(X)&=&\{f:X\rightarrow{\mathbb R}\,;\,f(x)+f(y)\geq d(x,y)\}.\\ E(X)&=&\{\textrm{minimal elements of the ordered set }\Delta(X)\}. \end{array}

Elements of {E(X)}, called by Isbell extremal functions, look like distance functions to set, from far away. The sup norm defined a metric on {E(X)}, in which {X} is isometrically embedded. {E(X)} is injective, it is the injective hull of {X}.

For finite sets {X}, {\Delta(X)} is a polyhedron, {E(X)} is a subcomplex of the boundary of {\Delta(X)}. For instance, for a 2-point space, {E(X)} is an interval. For a 3-point space, {E(X)} is a tripod. For a 4-point space, {E(X)} is a rectangle modelled on {\ell_{\infty}} with 4 antennas attached. For 5-point spaces, several combinatorial types occur, again consisting of {\ell_{\infty}}-rectangles and antennas. Some are planar, others are not.

3.3. Combinatorial dimension

Definition 6 (Dress, 1984) Let {X} be a metric space. The combinatorial dimension of {X} is the supremum of the dimensions of {E(S)}, {S} a finite subset of {X}.

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 {X} has finite combinatorial dimension, then for all {x_0,y_0 \in X}, there exists {\delta>0} such that

\displaystyle  \begin{array}{rcl}  d(x_0,y_0)+d(x,y)\leq d(x,y_0)+d(x_0,y) \end{array}

for all {x\in B(x_0,\delta)}, {y\in B(y_0,\delta)}.

This clearly fails for Euclidean plane, but holds for the line.

3.4. Proof of Theorem 2

Consider {h(s,t)=d(\sigma(s),\sigma(t))}, {\sigma=\sigma_{xy}}. 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.

3.5. Application

Theorem 8 Let {\Gamma} be a hyperbolic group, equipped with some word metric. Then {E(\Gamma)} is a proper finite dimensional polyhedral complex with finitely many isometry types of cells. {\Gamma} acts properly and cocompactly on {E(\Gamma)}.

It follows that {E(\Gamma)} has a unique {\Gamma}-invariant convex and consistent geodesic bicombing. This is weaker that {CAT(0)}, but not that bad.

3.6. Questions

What if one changes the metric on cells to a Euclidean metric, does it improve ? In dimension 2, this works well and produces a {CAT(0)} space. In higher dimensions, it may work. In progress.

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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