## Notes of Urs Lang’s Cambridge lecture 11-01-2017

Groups acting on spaces with a distinguished geodesic structure

Survey of work of my students and myself over the last few years.

1. Bicombings

A bicombing of a metric space ${X}$ is a family of paths ${\sigma_{xy}}$ joining pairs of points ${x}$ and ${y}$, such that

$\displaystyle \begin{array}{rcl} d(\sigma_{xy}(t),\sigma_{xy}(t'))\leq(1-t)d(x,x')+td(y,y'). \end{array}$

Say a bicombing is consistent if ${\sigma_{x'y'}\subset im \sigma_{xy}}$ if ${x', y'\in im \sigma_{xy}}$. This implies that

$\displaystyle \begin{array}{rcl} t\mapsto d(\sigma_{xy}(t),\sigma_{x'y'}(t)) \end{array}$

is convex.

1.1. Examples

1. Convex subsets of normed spaces.
2. If ${\rho:\bar X\rightarrow X}$ is a 1-Lipschitz retraction, and ${\bar \sigma}$ is a bicombing of ${\bar X}$, then ${\rho\circ\sigma_{|X\times X}}$ is a bicombing of ${X}$.
3. As a consequence, every absolute 1-Lipschitz retract has a bicombings. Such spaces are also called injective metric spaces. In fact, every injective metric spaces has an equivariant bicombing. There are plenty of injective metric spaces: every metric space embeds isometrically into an injective space, its injective hull (Isbell 1964).
4. There exist nonconvex bicombings. For instance, let ${X}$ be the subset of ${({\mathbb R}^2,|.|_\infty)}$ consisting of the ${\ell_1}$ unit ball union two wings. The vertical retraction is 1-Lipschitz. The image bicombining is not convex.
5. (Basso-Miesch 2016). There are convex, nonconsistent bicombings.

One can keep in mind the hierarchy

$\displaystyle \begin{array}{rcl} CAT(0)\Rightarrow\textrm{ Busemann }\Rightarrow \textrm{ consistent bicombing }\Rightarrow \textrm{ convex bicombing }\Rightarrow \textrm{ bicombing }. \end{array}$

1.2. Improving bicombings

Basso-Miesch: if ${X}$ is complete and ${\tilde\sigma}$ is a bicombing, there is a reversible bicombing, meaning that ${\sigma_{xy}(t)=\sigma_{yx}(1-t)}$. It is obtained by a limiting process

Theorem 1 (Descombes-Lang 2015) If ${X}$ is proper, a bicombing can be upgraded to a convex bicombing.

Theorem 2 Assume that ${X}$ has finite combinatorial dimension (sup of dimensions of injective hulls of finite subspaces). If ${X}$ admits a bicombing, it is unique, equivariant, reversible and consistent.

Corollary 3 Every hyperbolic acts geometrically on a proper metric space with a consistent, reversible, equivariant bicombing.

1.3. Barycenters

Theorem 4 (Es Sahib-Heinich 1999, Navas 2013 for Busemann spaces) Assume ${X}$ is complete, with a reversible bicombing. Then there exist a barycenter satisfying a sharp estimate: ${\forall n}$, ${\exists b_n:X^n\rightarrow X}$ such that

1. ${b_n(x_1,\ldots,x_n)\in\overline{conv_\sigma(\{x_1,\ldots,x_n\})}}$.
2. $\displaystyle d(b_n(\mathbf{x}),b_n(\mathbf{y}))\leq\frac{1}{n}\min_{\pi\in\mathfrak{S}_n}\sum_{i=1}^n d(x_i,y_{\pi(i)}).$

3. Isometry equivariance.

This can be extended to a barycenter for probability measures with finite first moment. This map is contracting in 1-Wasserstein distance.

Conversely, if a metric space ${X}$ admits a barycenter map which is 1-Lipschitz with respect to ${W_1}$, then ${\sigma_{xy}(t)=bar((1-t)\delta_x+t\delta_y}$ defines a bicombing.

1.4. Fixed points

Example (Basso 2016). There is a complete bounded Busemann space with a fixed-point free isometry.

${\ell_1({\mathbb Z})}$ admits an equivalent norm which is strictly convex. The closed convex hull of the orbit of ${\delta_0}$ under the shift is our space.

However, as soon as there exists an invariant compact set, there must be a fixed point.

Say an isometry ${p}$ is hyperbolic if ${\inf d(x,px)}$ is attained. Then ${p}$ has an axis (use the barycenter of a segment of orbit of a minimizer).

1.5. Flat subspaces

Theorem 5 (Descombes-Lang 2016) Let ${X}$ have a consistent bicombing. Two parallel ${\sigma}$-lines bound a strip isometric to a strip in a normed plane (the strip need not be unique).

Theorem 6 (Bowditch for Busemann spaces, Descombes-Lang 2016) Let ${X}$ have a consistent ${\Gamma}$-equivariant bicombing, where ${\Gamma}$ acts geometrically. Then

1. Every isometry ${p\in\Gamma}$ has a fixed point or a ${\sigma}$-axis.
2. If ${{\mathbb Z}^d\subset\Gamma}$, then ${X}$ contains a (normed) flat ${d}$-plane on which ${{\mathbb Z}^d}$ acts by translation.