Infinite flat rank of projective filling geodesic currents

Establish whether the flat rank and geometric rank of the space of projective filling geodesic currents equipped with the Lipschitz metric are both infinite.

Background

The paper proves that the geometric rank and asymptotic dimension of the space of projective filling geodesic currents are infinite. However, its construction uses finite simplices whose flat rank is uniformly bounded, so it does not produce quasi-isometrically embedded Euclidean orthants of arbitrarily large dimension.

The authors have only established that the flat rank is at least 8g−9 and explicitly leave open whether it is infinite, or whether it agrees with the already-proved infinite geometric rank.

References

We have not determined the flat rank of projective filling geodesic currents, although we know it is at least $8g-9$ by Theorem~\ref{mainthm:max_rank}.

— 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 “Infinite rank and dimension”

Can we compute or bound the several notions of dimension growth of $(P(S),d_{)$?

— 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 “Infinite rank and dimension”

What are $rank(\operatorname{Hit}n(S),d{)$ and $rankF(\operatorname{Hit}n(S),d{)$?

— 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 “Flats from Hitchin representations”