L1 distortion of higher-dimensional finite lamplighter groups

Determine whether every embedding of the finite lamplighter group \(_2\wr_n^m\) into \(L_1\), for integers \(m\ge 2\), incurs distortion at least a positive universal constant multiple of \(m\log n\).

Background

The planar case establishes a logarithmic lower bound for the Euclidean distortion of the square-root word metric and yields corresponding information about L1 embeddings. Motivated by higher-dimensional work using random measures on dyadic cubes, the authors formulate a broader question for lamplighter groups over the m-dimensional discrete torus. They state that the proposed lower bound would match the known upper bound.