Non-Bilipschitz Nets and Lattices in Amenable Groups

Definitions and mathematical statement

Let $\Gamma$ be a finitely generated group, let $S$ be a finite symmetric generating set, and let $d_S(g,h)$ be the minimum length of a word in $S$ representing $g^{-1}h$. The group is amenable if, for every finite $K\subseteq\Gamma$ and every $\varepsilon>0$, there exists a nonempty finite $F\subseteq\Gamma$ such that $|kF\mathbin{\triangle}F|<\varepsilon|F|$ for every $k\in K$, where $\triangle$ denotes symmetric difference.

A separated, coarsely dense net in $(\Gamma,d_S)$ means a subset $N\subseteq\Gamma$ for which there exist $r>0$ and $R<\infty$ such that distinct points of $N$ have distance at least $r$, and every point of $\Gamma$ has distance at most $R$ from some point of $N$. Give $N$ the restricted metric. Separation is automatic for subsets of a group with a word metric; this definition makes explicit the conventional meaning of "net."

Metric spaces $(X,d_X)$ and $(Y,d_Y)$ are bi-Lipschitz equivalent, written $X\simeq_{\mathrm{biLip}}Y$, if there exist a bijection $f:X\to Y$ and a constant $L\geq1$ such that \[ L^{-1}d_X(x,x')\leq d_Y(f(x),f(x'))\leq Ld_X(x,x')\qquad(x,x'\in X). \] The bijection need not respect any group structure.

Question 1, with the supplied quantifier retained. Is it true that \[ \forall\,\Gamma\text{ finitely generated and amenable},\quad \exists\,N\subseteq\Gamma\text{ a separated, coarsely dense net} \quad N\not\simeq_{\mathrm{biLip}}\Gamma? \] This formulation is independent of the finite generating set, since word metrics associated with finite generating sets are bi-Lipschitz equivalent. The supplied record imposes no infinitude or nontriviality hypothesis on $\Gamma$, so finite groups, including the trivial group, remain within its literal scope. Whether such groups were intended to be excluded is unresolved; no exclusion is inserted here.

A cocompact lattice in a Lie group $G$ means a discrete subgroup $\Lambda\leq G$ for which $G/\Lambda$ is compact. Write $\mathfrak g$ for the Lie algebra of $G$. Under the Lie-algebra convention, solvability means that the series $\mathfrak g^{(0)}=\mathfrak g$, $\mathfrak g^{(j+1)}=[\mathfrak g^{(j)},\mathfrak g^{(j)}]$ eventually vanishes; nilpotence means that $\mathfrak g_1=\mathfrak g$, $\mathfrak g_{j+1}=[\mathfrak g,\mathfrak g_j]$ eventually vanishes. For disconnected Lie groups, these conditions need not express solvability or nilpotence of the entire group, so the intended convention requires clarification.

Question 2, retaining its unresolved metric qualifications. Does there exist a solvable or nilpotent Lie group $G$ containing cocompact lattices $\Lambda_1,\Lambda_2$ such that \[ \Lambda_1\not\simeq_{\mathrm{biLip}}\Lambda_2? \] Here the lattices are to be compared as metric spaces, but the record does not specify their metrics. If $G$ is connected, a customary interpretation uses the restrictions of the distance of a left-invariant Riemannian metric on $G$; the existence of a bi-Lipschitz equivalence is independent of the choice of such a Riemannian metric. This is an explicitly conditional interpretation: connectedness, the treatment of disconnected groups, and the intended lattice metric convention are not supplied. Thus Question 2 does not yet determine a fully specified metric statement without these qualifications being resolved.

Short English statement

The two supplied questions ask whether every finitely generated amenable group contains a coarsely dense net that is not bi-Lipschitz equivalent to the group, and whether one solvable or nilpotent Lie group can contain two cocompact lattices that are not bi-Lipschitz equivalent. The first question leaves finite groups in scope; the second leaves connectedness and metric conventions unspecified. These qualifications require human mathematical review.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, with catalog/problems.json as the additional-record registry; permanent ID $30003780$, title "Non-Bilipschitz Nets and Lattices in Amenable Groups," exact source code catalogue entry. The supplied collection is "Oberwolfach Reports Open Problems," and the proposed year is $2018$. No individual proposer is identified.
• Supplied report citation: Mini-Workshop: Superexpanders and Their Coarse Geometry (2019), Oberwolfach Reports, pp. 1117--1160, DOI 10.4171/owr/2018/19. https://doi.org/10.4171/owr/2018/19. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The literature-source label calls this the "OWR 2018 report"; the relationship between that label, the proposed year, and the cited 2019 date has not been independently checked.
• Supplied literature source: Dymarz, "quasi-isometric versus biLipschitz." https://arxiv.org/abs/0904.3764. This is the supplied label; no fuller bibliographic identification or claim of applicability is added.
• Literature assessment checked at 2026-08-21: the supplied dated curation labels the problem open and states that non-bi-Lipschitz separated nets are known in many amenable, nilpotent, and solvable examples, while the universal amenable-group assertion and the two-cocompact-lattices question remain open. The background explicitly describes this as dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims. This assessment is supplied dated curation, not independently verified here.
• The record labels its statement corrected_verified and says that its mathematical content was preserved while notation, typography, and omitted definitions were repaired for self-contained publication. These are supplied editorial assertions, not a mathematical verification of this draft or of the unresolved qualifications above.