Sublinear Bilipschitz Equivalence and the Quasiisometric Classification of Solvable Lie Groups

We prove a product theorem for sublinear bilipschitz equivalences which generalizes the classical work of Kapovich, Kleiner and Leeb on quasiisometries between product spaces. We employ our product theorem to distinguish up to quasiisometry certain families of solvable groups which share the same dimension, cone-dimension and Dehn function; actually we do this by distinguishing them up to sublinear bilipschitz equivalence, which is slightly stronger. As an application, we recover the fact, recently obtained by Bourdon and Rémy with different groups, that there exists uncountably many quasiisometry classes of indecomposable, non-unimodular, high rank solvable Lie groups.

💡 Research Summary

The paper develops a product theorem for sublinear bilipschitz equivalences (SBEs), extending the classical Kapovich‑Kleiner‑Leeb (KKL) results on quasi‑isometries of product spaces. An SBE is an O(u)‑bilipschitz map where u(r)/r → 0; two maps are O(u)‑close if their pointwise distance is bounded by a function comparable to u(|x|). The authors prove that if X = M × ∏{i=1}^n X_i and Y = N × ∏{j=1}^m Y_j, where each factor is of coarse type I (negatively curved manifolds) or II (irreducible higher‑rank symmetric spaces), and there exists an O(u)‑SBE φ : X → Y, then the Euclidean factors have the same dimension (p = q) and the numbers of non‑Euclidean factors coincide (n = m). Moreover, after a suitable re‑indexing σ, there are O(u)‑SBEs φ_i : X_i → Y_{σ(i)} such that φ is O(v)‑close to the product map Φ = (φ_1,…,φ_n) for some sublinear v. This is Theorem A. Theorem C shows that if, in addition, the induced map on every u‑admissible asymptotic cone preserves the product structure, then the same factor‑wise decomposition holds directly in the original spaces, without passing to cones. The proof relies on the KKL topological splitting theorem (Theorem 5.1 in KKL) and careful control of sublinear error terms; the authors avoid the “non‑translatability” hypothesis used in KKL by imposing subadditivity and monotonicity on the error functions.

The second major component is the application to completely solvable Lie groups via Cornulier’s ρ₁‑reduction. For a completely solvable group S, let R exp S be the largest normal nilpotent subgroup; the group ρ₁(S) = R exp S ⋊ α_σ(S/R exp S) belongs to class (C₁) and is O(log)‑SBE to S (Cornulier 2011). Theorem D states that if two completely solvable groups S and S′ are O(u)‑SBE (in particular quasi‑isometric), and their ρ₁‑reductions split as ρ₁(S) ≅ ℝⁿ × P × H₁ × … × H_m, ρ₁(S′) ≅ ℝ^{n′} × P′ × H′₁ × … × H′_{m′}, where P, P′ are maximal completely solvable subgroups of semisimple groups (AN‑type) and each H_i, H′_j is a negatively curved Heintze group with abelian derived subgroup, then the two decompositions are isomorphic: n = n′, m = m′, P ≅ P′, and after re‑indexing H_i ≅ H′_i. Consequently, the original groups S and S′ are not quasi‑isometric unless their ρ₁‑structures match.

The authors illustrate the power of this machinery with several examples. In dimension four, they distinguish the groups G₀⁴,⁹ and ℝ × G_α³,⁵ (α∈(0,1)) which share dimension, cone‑dimension, and Dehn function, but have different ρ₁‑splittings, proving they are not quasi‑isometric. In dimension five, they recover a recent result of Bourdon‑Rémy: there exist uncountably many quasi‑isometry classes of indecomposable, non‑unimodular, completely solvable Lie groups with quadratic Dehn function. Their families (G_{1,β}^{5,19} for β>0) differ from the Bourdon‑Rémy families (G_{α,1‑α}^{5,33} for α≥½) by the presence or absence of a CAT(0) left‑invariant metric, which is detected via the ρ₁‑splitting and the SBE framework.

Overall, the paper introduces a robust “sublinear bilipschitz” perspective on large‑scale geometry, showing that product structures are rigid not only under quasi‑isometries but also under much weaker sublinear distortions. By coupling this with Cornulier’s reduction, the authors obtain new quasi‑isometric invariants for solvable Lie groups, resolve several classification problems that were previously inaccessible, and provide a unified approach that connects asymptotic cones, L^p‑cohomology, Dehn functions, and curvature properties. The results open avenues for further exploration of sublinear rigidity phenomena in other classes of groups and metric spaces.