Bilipschitz Group Equivalences and 3-Manifold Hyperbolicity

Definitions and mathematical statement

For metric spaces $(Y,d_Y)$ and $(Z,d_Z)$, a bi-Lipschitz bijection is a bijection $f:Y\to Z$ such that for some $C\geq1$, \[ C^{-1}d_Y(x,y)\leq d_Z(f(x),f(y))\leq C d_Y(x,y)\quad(x,y\in Y). \] For finitely generated groups, use word metrics from finite generating sets; their bi-Lipschitz class does not depend on the finite generating set. Equip a product with ${\mathbb{Z}}^2$ with the corresponding product word metric. No map is required to be a homomorphism and the constant may depend on the groups.

The source bundles four questions:

1. For every finitely generated group $G$, is there a bi-Lipschitz bijection $G\to G\times{\mathbb{Z}}^2$?
2. For every finite-index subgroup $G_1\leq G$ of a finitely generated group, is there a bi-Lipschitz bijection $G_1\to G$?
3. For two uniform lattices $L_1,L_2$ in the same Lie group, is there a bi-Lipschitz bijection $L_1\to L_2$? A uniform lattice is a discrete cocompact subgroup. Word metrics apply when the lattices are finitely generated; the source does not give the Lie-group hypotheses or an alternate metric convention otherwise.
4. If a compact three-manifold $M$ admits a partially hyperbolic diffeomorphism with invariant splitting \[ TM=E^s\oplus E^c\oplus E^u,\qquad \dim E^s=\dim E^c=\dim E^u=1, \] what restrictions follow for the topology of $M$? Conventionally, partial hyperbolicity means uniform contraction on $E^s$, uniform expansion on $E^u$, and domination of the center by the unstable and of the stable by the center, after suitable iterates and choice of metric.

The first literal quantifier includes finite groups, for which there cannot be a bijection to $G\times{\mathbb{Z}}^2$ on cardinality grounds. The second likewise needs finite cases checked if $G_1$ is proper. These are formulation obstructions to treating the supplied unrestricted wording as a verified open universal assertion. No replacement stabilization, exclusion of groups, or extra hypothesis is guessed. Quasi-isometry is not substituted for bi-Lipschitz bijectivity.

The final topology question is a separate broad task, not a consequence of the preceding group questions. The source does not specify closed versus boundary-bearing manifolds, orientability, diffeomorphism regularity, precise domination convention or the type of topological characterization sought.

The supplied triage checked 2026-08-21 labels the bundle open and says the stabilization, finite-index and broad topology questions remain unknown in recent literature. That claim is retained as attributed curation, with the literal finite-case issue explicitly flagged rather than accepted as a mathematical verification.

Short English statement

Preserve three proposed bi-Lipschitz group comparisons and the separate topology question for three-dimensional partially hyperbolic dynamics. The universal group wording has finite-case obstructions, and the lattice metrics and dynamical conventions need clarification.

Sources

• Local UnsolvedMath registry, permanent ID 30001042, source code catalogue entry; supplied complete record and corrected-verified notation-cleanup note. Literature checked 2026-08-21: the assessment is supplied dated curation, not independently verified here.
• Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/42, pp. 2337--2406: https://doi.org/10.4171/owr/2008/42. Proposed-year metadata gives 2008.
• Petrunin, metric geometry monograph: https://amathr.org/wp-content/uploads/2025/01/Petrunin_AMRResMonograghs2Ed2Pr.pdf.