Asymptotic dimension of Teichmüller space with the Lipschitz metric

Determine an upper bound, in terms of the genus g, for the asymptotic dimension of Teichmüller space equipped with the Lipschitz metric, and determine whether this asymptotic dimension is finite.

Background

The paper proves that the geometric and flat ranks of Teichmüller space with the Lipschitz metric are both equal to 3g−3. It does not determine the corresponding asymptotic dimension, which is only known to be at least 3g−3 by the rank lower bound.

The authors note that the hierarchical-hyperbolicity machinery used to obtain asymptotic-dimension bounds for Teichmüller space with other metrics does not currently apply to the Lipschitz metric. Thus, even finiteness of the asymptotic dimension remains unresolved.

References

At the moment we do not even know if $\adim(\Teich(S),d_{)$ is finite.

— Large-scale rank and geometry of the Lipschitz metric on geodesic currents  (2610.03359 - Martínez-Granado et al., 2 Oct 2026) in Section “Further questions,” paragraph “Asymptotic dimension of Teich(S)”