Planar lamplighter is not of negative type

Abstract: The lamplighter group over the planar integer grid is proved to not be bi-Lipschitz equivalent to any metric space of negative type, so in particular it does not admit a bi-Lipschitz embedding into $L_1$. This shows the existence of finitely generated metabelian groups on which word metrics are never comparable up to constant factors to conditionally negative definite (CND) kernels, and that the property of admitting a word metric-comparable CND kernel is not preserved by wreath products.

• The paper proves that the planar lamplighter group \(\mathbb{Z}_2 \wr \mathbb{Z}^2\) is not CND-comparable, with Euclidean distortion \(\asymp \sqrt{\log n}\) for its finite quotients under the square-root metric.
• A quadratic Poincaré inequality comparing Bernoulli noise with randomly sparsified crosses, combined with expected traveling-salesman estimates, yields the lower bound and resolves the Naor–Peres problem on bi-Lipschitz embeddings into \(L_1\).
• The results show that wreath products need not preserve CND-comparability, establish sharp \(L_q\) snowflake distortion for \(1\le q\le2\), and determine compression exponents for zero sections over polynomial-growth groups.

Overview and main result

The paper proves that the planar lamplighter group $\mathbb{Z}_2 \wr \mathbb{Z}^2$, equipped with any word metric, is not bi-Lipschitz equivalent to a metric of negative type. Concretely, the main theorem states that $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$ does not embed into $\ell_2$ with finite distortion, and more precisely that for every $n \ge 2$ the Euclidean distortion of $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$ is of order $\sqrt{\log n}$ (2608.16706). Since Schoenberg's classical characterization identifies metrics whose square roots embed isometrically into Hilbert space with conditionally negative definite (CND) kernels, the theorem resolves a problem posed by Naor–Peres: $\mathbb{Z}_2 \wr \mathbb{Z}^2$ admits no bi-Lipschitz embedding into $L_1$.

The result carries three structural corollaries, each stated with its justification in the introduction. First, $\mathbb{Z}_2 \wr \mathbb{Z}^2$ is the first metabelian (solvable of derived length two) group that fails to be CND-comparable; this is optimal in the derived-length hierarchy since finitely generated Abelian groups are CND-comparable. It is also the first "classical" group known to lack this property — prior examples were groups constructed specifically for the purpose, with derived length at least three or non-amenability. Second, the theorem separates CND-comparability from attaining Hilbert compression exponent $1/2$: Naor–Peres showed that $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$0 has Hilbert compression exponent exactly $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$1 and that the supremum is attained, yet the group is not CND-comparable. Third, while wreath products are known to preserve the Haagerup property, coarse Hilbert embeddability, and positive compression exponent, they do not preserve CND-comparability.

The proof also yields quantitative lower bounds on distortion growth: $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$2 for $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$3-point subsets, and the authors conjecture this is sharp, i.e., $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$4.

The quadratic cross inequality

The technical core is a Poincaré-type inequality for functions on the Boolean cube $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$5. Let $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$6 be $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$7-Bernoulli noise on $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$8 and let $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$9 be the indicator of a random sparsification of an axis-parallel cross $\ell_2$0 with uniformly random center $\ell_2$1, each cross element retained independently with probability $\ell_2$2. The key proposition asserts that every $\ell_2$3 satisfies

$\ell_2$4

Because the inequality is quadratic and the Walsh characters form an orthonormal basis of $\ell_2$5, it suffices to verify it on characters. For $\ell_2$6, the left side evaluates via $\ell_2$7 to a quantity of order $\ell_2$8, while the right side equals $\ell_2$9 using $n \ge 2$0. The authors note the implicit constant can be taken as $n \ge 2$1 and that this is optimal. The use of a single randomly shifted, randomly sparsified cross simplifies the multi-scale union of crosses employed in the antecedent work of Gartland–Ostrovskii.

From the inequality to the lower bound

Assuming an embedding $n \ge 2$2 of $n \ge 2$3 into $n \ge 2$4 with distortion $n \ge 2$5, one restricts attention to the zero section $n \ge 2$6 via $n \ge 2$7 and substitutes into the cross inequality. Two traveling-salesman estimates then drive the argument:

$n \ge 2$8

The upper bound follows from an explicit $n \ge 2$9 tour of the cross; the lower bound follows from a general lemma bounding expected TSP cost of Bernoulli subsets via volume growth, instantiated through a proposition showing that for any finite group of polynomial growth degree $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$0 with nondegenerate diameter, $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$1 when $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$2. Combining these gives $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$3, hence $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$4, matching the known upper bound from Baudier–Motakis–Schlumprecht–Žák. The passage from the finite quotients to the infinite group uses a lemma establishing mutual constant-distortion compatibility between the zero section of $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$5 and lamp configurations supported on $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$6; the nontrivial direction relies on a combinatorial argument handling "transitions" across the boundary of the fundamental box. A remark extends the non-CND-comparability conclusion to $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$7 for arbitrary infinite finitely generated $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$8.

Sharpness of the snowflake constants in $(\mathbb{Z}_2 \wr \mathbb{Z}^2_n, d^{1/2})$9

The same machinery yields an optimal refinement of the Naor–Peres compression theory. For $\sqrt{\log n}$0, let $\sqrt{\log n}$1 be the best constant in a lower Snowflake bound $\sqrt{\log n}$2 for $\sqrt{\log n}$3-Lipschitz maps into $\sqrt{\log n}$4. The paper proves $\sqrt{\log n}$5, sharpening the previously known lower bound $\sqrt{\log n}$6 obtained by tracking constants in Naor–Peres. The deduction composes a hypothetical map with the Schoenberg embedding $\sqrt{\log n}$7 satisfying $\sqrt{\log n}$8, applies the cross inequality, and optimizes over the scale parameter $\sqrt{\log n}$9, choosing $\mathbb{Z}_2 \wr \mathbb{Z}^2$0. More generally, the paper establishes

$\mathbb{Z}_2 \wr \mathbb{Z}^2$1

with the upper bound following from the $\mathbb{Z}_2 \wr \mathbb{Z}^2$2 embedding of Baudier et al. composed with the Bretagnolle–Dacunha-Castelle–Krivine embedding $\mathbb{Z}_2 \wr \mathbb{Z}^2$3.

Compression exponents of higher-dimensional zero sections

Independently of the negative-type question, the paper determines the $\mathbb{Z}_2 \wr \mathbb{Z}^2$4 compression exponent of the zero section of $\mathbb{Z}_2 \wr \mathbb{Z}^2$5 whenever $\mathbb{Z}_2 \wr \mathbb{Z}^2$6 has polynomial growth of degree $\mathbb{Z}_2 \wr \mathbb{Z}^2$7: for all $\mathbb{Z}_2 \wr \mathbb{Z}^2$8 and $\mathbb{Z}_2 \wr \mathbb{Z}^2$9,

$L_1$0

which recovers the Naor–Peres value $L_1$1 at $L_1$2 and extends it to all dimensions. The upper bound embeds rescaled Hamming cubes of dimension $L_1$3 into the zero section — constructed via witnesses to failure of TSP-efficiency in the sense of Gartland–Randrianantoanina–Randrianarivony — and applies Enflo-type rounding inequalities. The matching lower bound constructs an explicit embedding into $L_1$4 built from "doubly local" snapshots $L_1$5, which encode multiscale structure through simultaneous ball-and-annulus intersections counted against nets $L_1$6; a multiplicity-two property of these snapshots and packing bounds on the nets control both Lipschitzness and the compression lower bound.

Limitations and open questions

The paper leaves several questions open. The $L_1$7 distortion estimate $L_1$8 is established only for $L_1$9; for $\mathbb{Z}_2 \wr \mathbb{Z}^2$0 even bounded distortion is unknown. Whether some snowflake $\mathbb{Z}_2 \wr \mathbb{Z}^2$1 with exponent $\mathbb{Z}_2 \wr \mathbb{Z}^2$2 embeds into Hilbert space remains open (the answer is negative for $\mathbb{Z}_2 \wr \mathbb{Z}^2$3). Relatedly, sketchability of the zero section — equivalently, sketchability of planar TSP tours under the standard connection — is unresolved, though a positive answer to the snowflake question would imply it. The conjectured sharpness $\mathbb{Z}_2 \wr \mathbb{Z}^2$4 is proved only as a lower bound. Finally, the authors ask whether embeddings of $\mathbb{Z}_2 \wr \mathbb{Z}^2$5 into $\mathbb{Z}_2 \wr \mathbb{Z}^2$6 must incur distortion $\mathbb{Z}_2 \wr \mathbb{Z}^2$7, which would be sharp by the known upper bounds; the methods here treat only $\mathbb{Z}_2 \wr \mathbb{Z}^2$8.

Conclusion

This paper settles the Naor–Peres problem by proving that the planar lamplighter group admits no bi-Lipschitz embedding into $\mathbb{Z}_2 \wr \mathbb{Z}^2$9, via a clean quadratic Poincaré inequality comparing Bernoulli noise to random crosses on the Boolean cube, combined with sharp expected-TSP estimates for Bernoulli subsets. The result identifies the first metabelian — indeed the first classical — group failing CND-comparability, demonstrates that wreath products need not preserve this property, and separates CND-comparability from attained Hilbert compression $1/2$0. The accompanying determination of compression exponents for zero sections of lamplighters over all polynomial-growth groups rounds out the quantitative picture, while several natural questions about higher dimensions, larger snowflake exponents, and sketchability remain open.