Metric Properness of Wreath-Product Actions on Median Graphs

Definitions and mathematical statement

A median graph is a connected graph $X$ such that, for every three vertices $x,y,z$, the intersection $I(x,y)\cap I(y,z)\cap I(z,x)$ consists of exactly one vertex, where $I(u,v)=\{w:d_X(u,v)=d_X(u,w)+d_X(w,v)\}$ and $d_X$ is the edge-path metric. Actions on graphs are understood to be by graph automorphisms. Following the source's convention, an action $G\curvearrowright X$ is proper if every vertex stabilizer is finite. It is metrically proper if, for a base vertex $o$, \[ \forall R\geq 0,\qquad \#\{g\in G:d_X(o,go)\leq R\}<\infty. \] This condition is independent of the base vertex. No local finiteness assumption on $X$ is supplied.

Let $N\trianglelefteq G$, put $Q=G/N$, and give $Q$ the left translation action of $G$. Define \[ A_Q=\bigoplus_{q\in Q}\mathbb Z/2\mathbb Z =\{f:Q\to\mathbb Z/2\mathbb Z:\operatorname{supp}(f)\text{ is finite}\}, \qquad (g\cdot f)(q)=f(g^{-1}q). \] The permutational wreath product in question is \[ W(G,N)=(\mathbb Z/2\mathbb Z)\wr_QG=A_Q\rtimes G, \qquad (f,g)(h,k)=(f+g\cdot h,gk). \] For any group $H$, write out the following property without assuming any additional restrictions on the median graph: \[ \text{there exist a median graph }Y\text{ and a metrically proper action }H\curvearrowright Y. \]

Question in the supplied original passage. Let $F_n$ be a free group of rank $n$, and let $N\trianglelefteq F_n$. Assume that $F_n/N$ has no metrically proper action on any median graph. Can $W(F_n,N)$ nevertheless admit a metrically proper action on a median graph? The passage names $F_n/N\simeq\operatorname{BS}(1,2)$ as an example of the quotient hypothesis, where \[ \operatorname{BS}(1,2)=\langle a,t\mid tat^{-1}=a^2\rangle. \] The assertion that this example satisfies the hypothesis is supplied source content, not independently verified here. The passage does not specify the range of $n$ or whether an existential example or a decision for every eligible normal subgroup is intended.

Broader formulation in the registry. Given a group $G$ admitting a proper action on a median graph and a normal subgroup $N$ such that $G/N$ has no metrically proper action on any median graph, determine whether $W(G,N)$ can admit a metrically proper action on a median graph, and characterize the pairs $(G,N)$ for which it does. This broadens the explicitly displayed free-group question. The word "upgraded" is interpreted here only as existence of some metrically proper median-graph action: no requirement to modify or retain a specified action is supplied.

Supplied context and qualifications. The passage describes a proper action of $W(G,N)$ on $A_Q\times X$ whenever $G\curvearrowright X$ is proper. Here $A_Q$ is regarded as the graph joining two finitely supported functions when they differ at exactly one coordinate, the product is the Cartesian graph product, and the indicated action is \[ (f,g)\cdot(h,x)=(f+g\cdot h,gx). \] This is the natural reconstruction of a typographically damaged product formula in the supplied original text.

The passage attributes to Chifan and Ioana the assertion that $W(G,N)$ is not a-T-menable whenever $G$ is a-T-menable but $G/N$ is not. Here a-T-menability means existence of a metrically proper affine isometric action on a Hilbert space, with metric properness expressed by finiteness of bounded-displacement sets. The passage also invokes a theorem of Niblo and Reeves to exclude metrically proper median-graph actions for the resulting wreath-product examples. Its ensuing free-group conclusion needs the preceding non-a-T-menability hypothesis on the quotient; it must not be read here as a conclusion for every $N$, or for every quotient lacking a metrically proper median-graph action. These are supplied background claims with incomplete theorem references and hypotheses requiring review.

The original passage additionally asks for a median-geometric understanding of these examples and mentions computing the "travelling polytope constant" of a product of regular trees. No definition of that constant, tree degrees, or number of factors is supplied, so this additional request cannot be made precise from the record alone.

Short English statement

Using the source's convention that properness means finite vertex stabilizers, ask whether a wreath product can have a metrically proper median-graph action even when its quotient indexing the lamps has none. The original explicit question concerns free groups; the registry extends it to general groups and asks for a characterization. The quantifier scope, meaning of "upgraded," and several background claims require review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006619$, source code catalogue entry; project registry data/unsolvedmath/problems.json, with catalog/problems.json as the additional-record catalog. The supplied original heading is "Problem 8. Metric actions on median graphs, Genevois." The structured proposer field is empty; the attribution to Genevois comes from that heading.
• Supplied report citation: Median Geometry and Applications (2026), Oberwolfach Reports, pp. 487--540, DOI 10.4171/owr/2026/8. https://doi.org/10.4171/owr/2026/8.
• Supplied literature source, labelled "OWR report": https://ems.press/content/serial-article-files/53603.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that no result settling the wreath-product case was found. This is supplied dated curation, not an independently verified literature assessment; the background expressly requests independent verification of time-sensitive status and cited claims.
• The original passage attributes background results to Chifan and Ioana, and to Niblo and Reeves, but supplies no theorem numbers or bibliographic references for those results.
• The registry labels the statement corrected_verified and says that only formula delimiters, notation, grammar, and typography were repaired, with content and scope preserved. These are supplied editorial labels, not evidence of independent mathematical verification.