Expanders Avoiding All CAT-Zero Spaces

Definitions and mathematical statement

The supplied question is: "Does there exist an expander that does not coarsely embed into any CAT(0) space?" Here is a precise formulation using the conventional interpretation of an expander as a family of finite graphs with their unscaled graph metrics. The supplied record does not explicitly specify this convention.

An expander family is a sequence $(G_n)_{n\geq 1}$ of finite connected simple undirected graphs $G_n=(V_n,E_n)$, with $|V_n|\to\infty$, for which there exist an integer $D\geq 1$ and a real number $h>0$ such that every vertex has degree at most $D$, and \[ |\partial_{E_n}A|\geq h|A| \quad\text{whenever }A\subseteq V_n,\quad 0<|A|\leq |V_n|/2. \] Here $\partial_{E_n}A$ consists of the edges with exactly one endpoint in $A$. Let $d_n$ be the shortest-path metric with every edge of length $1$.

Represent the family by the metric space $X=\bigsqcup_{n\geq1}V_n$. For definiteness, write $\Delta_n=\operatorname{diam}(V_n,d_n)$ and set \[ d_X(x,y)= \begin{cases} d_n(x,y),&x,y\in V_n,\\ \Delta_n+\Delta_m+n+m,&x\in V_n,\ y\in V_m,\ n\neq m. \end{cases} \] This is a coarse disjoint union metric: it retains each graph metric and separates distinct components at distances tending to infinity. The displayed choice fixes the otherwise unspecified distances between components.

A metric space $Y$ is $\operatorname{CAT}(0)$ if it is geodesic and every geodesic triangle satisfies Euclidean comparison: distances between points on its sides are no greater than the distances between the corresponding points on the sides of a Euclidean triangle with the same side lengths. Completeness is not imposed here. Requiring complete $\operatorname{CAT}(0)$ targets gives the same existence question, since the metric completion of a $\operatorname{CAT}(0)$ space is again $\operatorname{CAT}(0)$.

A map $f:X\to Y$ is a coarse embedding if there exist nondecreasing functions $\rho_-,\rho_+:[0,\infty)\to[0,\infty)$, with $\lim_{t\to\infty}\rho_-(t)=\infty$, such that \[ \rho_-(d_X(x,x'))\leq d_Y(f(x),f(x'))\leq\rho_+(d_X(x,x')) \qquad(x,x'\in X). \]

Question. Does there exist an expander family as above such that, for every $\operatorname{CAT}(0)$ metric space $Y$, there are no map $f:X\to Y$ and control functions $\rho_-,\rho_+$ satisfying these conditions? The same family must obstruct every such target; the family cannot depend on $Y$.

The graph-family and metric conventions above are an explicit interpretation requiring comparison with the original report. The supplied registry labels the problem open; this label and its literature assessment are supplied curation, not an independently verified mathematical conclusion.

Short English statement

Is there one family of finite connected graphs, with uniformly bounded degrees and uniformly positive edge expansion, whose coarse disjoint union cannot coarsely embed into any $\operatorname{CAT}(0)$ metric space? Graph distances are unscaled, and the same family must work for every target. These conventions make explicit the otherwise unspecified meaning of "an expander" in the supplied question.

Sources

• Local UnsolvedMath registry, supplied complete record for permanent ID $30003779$, source problem code catalogue entry, title "Expanders Avoiding All CAT-Zero Spaces," in the Oberwolfach Reports Open Problems collection. The record labels the statement exact and its status open, and gives proposed year $2018$; no individual proposer is supplied. These are registry assertions, not independent verification.
• Supplied report citation: Mini-Workshop: Superexpanders and Their Coarse Geometry (2019), Oberwolfach Reports, DOI $10.4171/owr/2018/19$, pp. 1117--1160. https://doi.org/10.4171/owr/2018/19. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions. The parenthetical year $2019$ differs from the registry's proposed year and year tag $2018$; the intended bibliographic dating has not been resolved.
• Supplied literature source, attributed to Mendel--Naor and labeled "expanders and Hadamard spaces": https://arxiv.org/abs/1306.5434.
• Supplied literature source labeled "OWR 2018 report": https://doi.org/10.4171/owr/2018/19.
• Literature assessment checked on 2026-08-21, according to the supplied record. This is supplied dated curation, not independently verified. It labels the problem open and states that Mendel--Naor construct expanders relative to substantial $\operatorname{CAT}(0)$ classes and discrete tori avoiding all $\operatorname{CAT}(0)$ spaces, while the existence of an expander avoiding every $\operatorname{CAT}(0)$ space remains open. The background expressly characterizes this as dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.