## 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 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.