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.


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
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