[Paper Review] The Euclidean distortion of the lamplighter group

Tim Austin, Assaf Naor+1 more|ArXiv.org|May 31, 2007

Mathematics and Applications12 references4 citations

TL;DR

This paper establishes that the Euclidean distortion of the cyclic lamplighter group $C_2 \wr C_n$ is $\Theta(\sqrt{\log n})$, matching the previously known lower bound. The authors construct an explicit, equivariant bi-Lipschitz embedding into Hilbert space using irreducible unitary representations, proving tightness of the distortion bound via representation-theoretic methods.

ABSTRACT

We show that the cyclic lamplighter group $C_2 \bwr C_n$ embeds into Hilbert space with distortion ${ m O}(\sqrt{\log n})$. This matches the lower bound proved by Lee, Naor and Peres in \cite{LeeNaoPer}, answering a question posed in that paper. Thus the Euclidean distortion of $C_2 \bwr C_n$ is $Θ(\sqrt{\log n})$. Our embedding is constructed explicitly in terms of the irreducible representations of the group. Since the optimal Euclidean embedding of a finite group can always be chosen to be equivariant, as shown by Aharoni, Maurey and Mityagin \cite{AhaMauMit} and by Gromov (see \cite{deCTesVal}), such representation-theoretic considerations suggest a general tool for obtaining upper and lower bounds on Euclidean embeddings of finite groups.

Motivation & Objective

• To close the gap between the known $\Omega(\sqrt{\log n})$ lower bound and the lack of matching upper bound for the Euclidean distortion of $C_2 \wr C_n$.
• To construct an explicit, low-distortion embedding of the lamplighter group into Hilbert space that achieves the optimal distortion.
• To demonstrate that representation-theoretic methods—specifically, the decomposition into irreducible unitary representations—can yield tight upper bounds on Euclidean distortion for finite groups.
• To confirm that the previously conjectured lower bound is tight, answering an open question from Lee, Naor, and Peres (2006).
• To explore the role of equivariant embeddings in minimizing distortion, building on results by Aharoni, Maurey, Mityagin, and Gromov.

Proposed method

• Construct a unitary representation of $C_2 \wr C_n$ as the direct sum of its irreducible representations.
• Select a specific vector $v$ in the Hilbert space such that the orbit map $g \mapsto \beta(g)v$ yields a bi-Lipschitz embedding into Hilbert space.
• Use the structure of the group's irreducible representations to control the distortion of the embedding via spectral and harmonic-analytic techniques.
• Leverage the fact that minimal distortion embeddings for amenable groups can be chosen to be equivariant, as established by Aharoni, Maurey, Mityagin, and Gromov.
• Analyze the distortion by bounding the ratio of Hilbert space distances to word metric distances using representation-theoretic norms and character sums.
• Apply known results on eigenvalues of Cayley graphs and zig-zag products to rule out better distortion bounds under certain generating sets.

Experimental results

Research questions

• RQ1 Is the $\Omega(\sqrt{\log n})$ lower bound on the Euclidean distortion of $C_2 \wr C_n$ tight?
• RQ2 Can an explicit, equivariant embedding into Hilbert space achieve distortion $O(\sqrt{\log n})$ for $C_2 \wr C_n$?
• RQ3 To what extent can the decomposition of a group's unitary representations be used to construct low-distortion embeddings into Hilbert space?
• RQ4 Does the choice of generating set significantly affect the Euclidean distortion of wreath products like $C_2 \wr C_n$?
• RQ5 Is the infimal distortion of $C_2 \wr C_n$ in $L^1$-space bounded or does it tend to infinity with $n$?

Key findings

• The Euclidean distortion of $C_2 \wr C_n$ is exactly $\Theta(\sqrt{\log n})$, confirming tightness of the previously known lower bound.
• An explicit, equivariant embedding into Hilbert space achieves distortion $O(\sqrt{\log n})$, matching the lower bound up to a universal constant.
• The embedding is constructed via the direct sum of irreducible unitary representations of the group, with a carefully chosen vector in the representation space.
• The construction demonstrates that representation-theoretic methods can yield sharp upper bounds on Euclidean distortion for finite groups.
• For a random generating set of size $\Omega(\log n)$, the distortion of $C_2 \wr C_n$ grows as $\Omega(\sqrt{n})$, showing strong dependence on the generating set.
• The result suggests that spectral methods may not always yield optimal bounds, and representation theory provides a more refined tool for analyzing distortion in non-Abelian groups.