Non-Helly Coarse-Helly Actions and Coarse-Median Artin Groups

Definitions and mathematical statement

Equip the vertex set of a connected graph $X$ with its combinatorial path metric $d_X$, and write $B_X(v,r)=\{w:d_X(v,w)\le r\}$, where $r\in{\mathbb{Z}}_{\ge0}$. Under the finite-family convention, $X$ is a Helly graph if every finite pairwise-intersecting family of vertex balls has nonempty total intersection. It is coarse Helly if there exists $\xi\in{\mathbb{Z}}_{\ge0}$ such that, for every $n\ge1$, vertices $v_1,\ldots,v_n$, and radii $r_1,\ldots,r_n\in{\mathbb{Z}}_{\ge0}$, \[ \left(\forall i,j,\ B_X(v_i,r_i)\cap B_X(v_j,r_j)\ne\varnothing\right) \Longrightarrow \bigcap_{i=1}^{n}B_X(v_i,r_i+\xi)\ne\varnothing. \] The constant $\xi$ is independent of the family. An action is called geometric if it is proper and cocompact by graph automorphisms, regarded as isometries of the graph. A Helly group is a group admitting a geometric action on some Helly graph. The supplied record does not specify the finite-family versus arbitrary-family convention for the Helly conditions, or further graph hypotheses; the definitions here are provisional conventions requiring source review.

Question 1 (group interpretation). Does there exist a group $G$ and a connected coarse-Helly graph $X$ such that $G$ acts geometrically on $X$, but $G$ is not a Helly group? Equivalently, is \[ \exists G\ \exists X\ \left[ X\text{ is coarse Helly},\quad G\curvearrowright X\text{ geometrically},\quad \nexists Y\text{ a Helly graph with }G\curvearrowright Y\text{ geometrically} \right] \] true? The background assessment supports this group interpretation. However, the wording "a coarse Helly graph that is not Helly" also admits the different interpretation that $X$ itself is not Helly. That alternative asks only for a geometric action on a coarse-Helly, non-Helly graph and does not require $G$ to fail to admit a geometric action on another Helly graph. The supplied record does not conclusively resolve this grammatical ambiguity.

For the second question, let $S$ be a finite set and let $M=(m_{st})_{s,t\in S}$ be symmetric, with $m_{ss}=1$ and $m_{st}\in\{2,3,\ldots\}\cup\{\infty\}$ for $s\ne t$. The associated Artin group is \[ A_M=\left\langle S\ \middle|\ \underbrace{sts\cdots}_{m_{st}\text{ letters}} = \underbrace{tst\cdots}_{m_{st}\text{ letters}} \text{ for }s\ne t\text{ with }m_{st}<\infty \right\rangle. \] There is no defining relation for a pair with $m_{st}=\infty$.

Here is the usual finite-approximation meaning of coarse median. A finite median graph $P$ is a finite connected graph such that, for each triple of vertices $a,b,c$, the intersection \[ I_P(a,b)\cap I_P(b,c)\cap I_P(c,a) \] consists of one vertex, denoted $m_P(a,b,c)$, where $I_P(a,b)=\{z:d_P(a,z)+d_P(z,b)=d_P(a,b)\}$. A metric space $(Z,d)$ is coarse median if there exist a map $\mu:Z^3\to Z$, a constant $K\ge0$, and a function $H:{\mathbb{Z}}_{\ge0}\to[0,\infty)$ satisfying both of the following conditions:

1. For all $x_i,y_i\in Z$, \[ d\left(\mu(x_1,x_2,x_3),\mu(y_1,y_2,y_3)\right) \le K\sum_{i=1}^{3}d(x_i,y_i)+H(0). \]
2. For every integer $p\ge1$ and every nonempty finite $F\subseteq Z$ with $|F|\le p$, there exist a finite median graph $P$ and maps $\pi:F\to V(P)$, $\lambda:V(P)\to Z$ such that \[ d(a,\lambda\pi(a))\le H(p)\qquad(a\in F) \] and \[ d\left(\lambda(m_P(u,v,w)),\mu(\lambda(u),\lambda(v),\lambda(w))\right) \le H(p)\qquad(u,v,w\in V(P)). \]

A finitely generated group is called coarse median if its word metric for a finite generating set is coarse median; this property is independent of the finite generating set. No uniform bound on the ranks of the finite median approximations is imposed here.

Question 2 (finite generating-set formulation). Characterize the finite Artin matrices $M$ for which $A_M$, equipped with a word metric, is coarse median. The source asks "which Artin groups" without explicitly restricting the size of $S$ or specifying a metric for infinitely generated groups. Thus the finite generating-set formulation above is a precise conventional formulation, but its agreement with the full intended scope requires review. No restriction to a particular class of Artin matrices is supplied.

Short English statement

The first question asks whether a group can act properly and cocompactly on a coarse-Helly graph while admitting no such action on any Helly graph. The source wording also permits a different reading concerning only the chosen graph. The second question asks which Artin groups are coarse median; the explicit formulation here covers Artin groups with finite defining generating sets. Both interpretations and the stated conventions require human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30004625$, source problem code catalogue entry, title Non-Helly Coarse-Helly Actions and Coarse-Median Artin Groups, in the Oberwolfach Reports Open Problems collection. The supplied record lists proposed year $2021$ and identifies no individual proposer.
• Supplied report 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 supplied background describes these reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or speaker attribution is supplied.
• Supplied literature reference: "Osajda--Valiunas, coarse Helly groups." https://arxiv.org/abs/2012.03246.
• Supplied literature reference: "Helly groups 2025." https://msp.org/gt/2025/29-1/gt-v29-n1-p01-s.pdf.
• Supplied literature reference: "Oberwolfach report." https://doi.org/10.4171/owr/2021/8.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the existence of a group acting geometrically on a coarse-Helly graph without being Helly is unsettled in general, and that the classification of coarse-median Artin groups remains incomplete. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The record expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied editorial claims and do not establish mathematical correctness or resolve the formulation issues recorded here.