Universality of Quasi-Isometry among Marked Groups

Definitions and mathematical statement

For $n\geq1$, let $F_n$ be the free group with its ordered free generators, and identify an $n$-marked group with a quotient $F_n/N$ for a normal subgroup $N\triangleleft F_n$. The space $\mathcal G_n$ of such normal subgroups is a closed subspace of $\{0,1\}^{F_n}$, equivalently carrying the topology of agreement on larger finite marked Cayley balls. Use the customary Polish coding \[ \mathcal G=\bigsqcup_{n\geq1}\mathcal G_n \] for finitely generated marked groups. Repeated or redundant generators are allowed by this quotient coding.

Equip each group with the word metric of its finite marking. Two groups $G,H$ are quasi-isometric if there exist a map $q:G\to H$ and constants $\lambda\geq1$, $C\geq0$ such that \[ \lambda^{-1}d_G(x,x')-C\leq d_H(q(x),q(x')) \leq\lambda d_G(x,x')+C \quad(x,x'\in G), \] and every point of $H$ lies within distance $C$ of $q(G)$. Write $E_{\mathrm{QI}}$ for this equivalence relation on $\mathcal G$; its definition concerns the groups with word metrics, rather than an isomorphism respecting the markings.

An equivalence relation $E\subseteq Y\times Y$ on a Polish space is $K_\sigma$ if it is a countable union of compact subsets of $Y\times Y$. A Borel reduction of $E$ to $E_{\mathrm{QI}}$ is a Borel map $r:Y\to\mathcal G$ such that \[ y\mathrel E y'\quad\Longleftrightarrow\quad r(y)\mathrel{E_{\mathrm{QI}}}r(y') \qquad(y,y'\in Y). \] The conjecture asks whether $E_{\mathrm{QI}}$ is a universal $K_\sigma$ Borel equivalence relation: it belongs to that class, and every $K_\sigma$ Borel equivalence relation on a Polish space has such a reduction to it. The reduction may depend on $E$, and it need not be continuous or injective. The supplied assessment reports the $K_\sigma$ upper bound as known; the target left open there is universality among finitely generated groups, not merely among bounded-degree graphs.

Short English statement

Is quasi-isometry of finitely generated marked groups as complicated under Borel reduction as every $K_\sigma$ Borel equivalence relation? The source reports the upper bound but conjectures the matching universality.

Sources

• Local UnsolvedMath registry, permanent ID $30000909$, source code catalogue entry, "Universality of Quasi-Isometry among Marked Groups." No individual proposer is supplied.
• Set Theory (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/02, pp. 79--118: https://doi.org/10.4171/owr/2008/02.
• Supplied reference, "Thomas": https://sites.math.rutgers.edu/~sthomas/quasi2.pdf.
• Supplied literature triage, checked 2026-08-21: status "open." It attributes to Thomas the $K_\sigma$ upper bound and a universal relation for connected four-regular graphs, while calling quasi-isometry universality for finitely generated groups conjectural. This dated assessment derives from supplied OWR curation; time-sensitive status and cited claims require independent verification.