Groups acting on spaces with a distinguished geodesic structure
Survey of work of my students and myself over the last few years.
A bicombing of a metric space is a family of paths joining pairs of points and , such that
Say a bicombing is consistent if if . This implies that
- Convex subsets of normed spaces.
- If is a 1-Lipschitz retraction, and is a bicombing of , then is a bicombing of .
- 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).
- There exist nonconvex bicombings. For instance, let be the subset of consisting of the unit ball union two wings. The vertical retraction is 1-Lipschitz. The image bicombining is not convex.
- (Basso-Miesch 2016). There are convex, nonconsistent bicombings.
One can keep in mind the hierarchy
1.2. Improving bicombings
Basso-Miesch: if is complete and is a bicombing, there is a reversible bicombing, meaning that . It is obtained by a limiting process
Theorem 1 (Descombes-Lang 2015) If is proper, a bicombing can be upgraded to a convex bicombing.
Theorem 2 Assume that has finite combinatorial dimension (sup of dimensions of injective hulls of finite subspaces). If 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.
Theorem 4 (Es Sahib-Heinich 1999, Navas 2013 for Busemann spaces) Assume is complete, with a reversible bicombing. Then there exist a barycenter satisfying a sharp estimate: , such that
- 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 admits a barycenter map which is 1-Lipschitz with respect to , then defines a bicombing.
1.4. Fixed points
Example (Basso 2016). There is a complete bounded Busemann space with a fixed-point free isometry.
admits an equivalent norm which is strictly convex. The closed convex hull of the orbit of 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 is hyperbolic if is attained. Then has an axis (use the barycenter of a segment of orbit of a minimizer).
1.5. Flat subspaces
Theorem 5 (Descombes-Lang 2016) Let have a consistent bicombing. Two parallel -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 have a consistent -equivariant bicombing, where acts geometrically. Then
- Every isometry has a fixed point or a -axis.
- If , then contains a (normed) flat -plane on which acts by translation.