Canonical Bicombings and Group Actions on Injective Spaces

Definitions and mathematical statement

A metric space $(X,d)$ is injective if, whenever $A$ is a metric subspace of a metric space $B$, every $1$-Lipschitz map $A\to X$ extends to a $1$-Lipschitz map $B\to X$. Write $E(X)$ for its injective hull; for injective $X$, identify $E(X)$ isometrically with $X$.

A geodesic bicombing on $X$ is a map $\sigma:X\times X\times[0,1]\to X$, with $\sigma_{xy}(t)=\sigma(x,y,t)$, such that \[ \sigma_{xy}(0)=x,\qquad \sigma_{xy}(1)=y,\qquad d(\sigma_{xy}(s),\sigma_{xy}(t))=|s-t|d(x,y). \] It is convex if, for every $x,y,x',y'\in X$, the function \[ t\longmapsto d(\sigma_{xy}(t),\sigma_{x'y'}(t)) \] is convex on $[0,1]$. It is consistent if, for every $x,y\in X$, $0\leq a\leq b\leq1$, and $u\in[0,1]$, \[ \sigma_{\sigma_{xy}(a),\sigma_{xy}(b)}(u)=\sigma_{xy}((1-u)a+ub). \] Reversibility, meaning $\sigma_{yx}(t)=\sigma_{xy}(1-t)$, is an additional condition and is not imposed here; the supplied record does not specify whether its bicombing convention includes it.

1. For every injective metric space $X$, the record refers to a canonical embedding and a retraction \[ j:X=E(X)\longrightarrow\Delta(X),\qquad r:\Delta(X)\longrightarrow E(X),\qquad r\circ j=\operatorname{id}_X. \] Is the bicombing induced by this construction convex? Is it consistent? These are separate questions. The supplied record does not define $\Delta(X)$, specify the retraction $r$, or give the rule producing the bicombing. Consequently it does not determine a particular bicombing, or whether any assertion is intended for one distinguished retraction or for every eligible retraction. This part remains a question about an incompletely specified construction.

2. Does there exist a compact metric space $(K,d)$ carrying two consistent geodesic bicombings $\sigma$ and $\tau$ such that \[ \exists x,y\in K\ \exists t\in[0,1]\qquad \sigma_{xy}(t)\ne\tau_{xy}(t)? \] The record imposes no additional convexity requirement. The reversibility convention remains subject to source confirmation.

3. Suppose a group $G$ acts geometrically on an injective metric space. Must $G$ be a Helly group? Must $G$ be a CAT(0) group? Must $G$ be biautomatic?

   For an explicit conventional reading of these terms, a geometric action of a discrete group is a proper, cocompact action by isometries, where proper means that $\{g\in G:gC\cap C\ne\varnothing\}$ is finite for every compact subset $C$. A Helly graph is a connected graph whose combinatorial closed balls have the finite Helly property: every finite pairwise intersecting family of such balls has nonempty total intersection. A Helly group admits a geometric action by graph automorphisms on a Helly graph. A CAT(0) group admits a geometric action on a CAT(0) geodesic metric space, whose geodesic triangles satisfy the Euclidean comparison inequality. A group is biautomatic if it has a finite symmetric generating set and a regular language of representative words satisfying uniform synchronous fellow-traveller bounds for multiplication by generators on both the left and the right.

   These definitions make the usual separate universal implications explicit. However, the supplied record does not state the precise group-action conventions, whether properness of the ambient space is additionally required, or whether its word "or" asks for the separate properties or merely their disjunction. These qualifications must be checked before treating this part as a uniquely specified conjecture.

Short English statement

The record asks whether a specified but incompletely described construction on injective spaces gives convex or consistent geodesics, whether a compact metric space can have two distinct consistent choices of geodesics, and which of the Helly, CAT(0), and biautomatic properties follow from a geometric action on an injective space. The construction and several conventions require source clarification. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Supplied local registry record, permanent UnsolvedMath ID $30004623$, title Canonical Bicombings and Group Actions on Injective Spaces, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2021.
• Supplied source citation: Nonpositively Curved Complexes (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/8, pp. 383--417. https://doi.org/10.4171/owr/2021/8. The background attributes the questions to workshop open-problem discussions. The supplied title year 2022 differs from the proposed year 2021 and the year appearing in the DOI; this discrepancy has not been resolved.
• Supplied literature source: XXL type Artin groups. https://www.numdam.org/item/10.5802/aif.3524.pdf. No authors or further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-22: the supplied status is open. The supplied assessment says that a compound list concerning Artin groups, injective hulls, Helly actions, and buildings has unresolved questions, with known classes and partial cases rather than answers to the full list. This is supplied dated curation derived from the OWR material, not independently verified literature research. Its broader scope does not identify additional precise questions in this record.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied editorial claims; they do not resolve the missing construction or constitute mathematical review of this draft. The cited sources have not been independently consulted here.