Distortion growth of finite planar lamplighter subsets in L1

Determine whether the largest possible L1 distortion of an n-point subset of the planar lamplighter group \(_2\wr^2\) satisfies \(c_1^n(_2\wr^2)\asymp \log\log n\) for every integer \(n\ge 3\).

Background

The paper proves the lower bound c1n(2≀2)≳log⁡log⁡nc_1^n(_2\wr^2)\gtrsim \log\log n for finite subsets of the planar lamplighter group. The authors explicitly ask whether this lower bound is sharp, noting their suspicion that the asserted asymptotic order is correct.

References

In terms of distortion growth, we get the following result and open question. For every integer n\ge 3, let c_1n(_2\wr2) denote the largest possible L_1 distortion of an n-point subset of _2\wr2. We prove herein that c_1n(_2\wr2)\gtrsim \log\log n, and ask if this is sharp (we suspect that it is): Is it true that c_1n\big(_2\wr2\big)\asymp \log\log n for every integer n\ge 3?

— Planar lamplighter is not of negative type  (2608.16706 - Antonelli et al., 17 Aug 2026) in Section 1, immediately after the discussion of the distortion growth bound