Quasi-isometry invariance of type FH_n(R)

Ascertain whether, for any normed ring R and integer n ≥ 1, the property of being of type FH_n(R) is preserved under quasi-isometry; specifically, determine whether every group H that is quasi-isometric to a group G of type FH_n(R) is itself of type FH_n(R).

Background

The paper notes that, unlike the case n = 2 where FP_2(R) is equivalent to FH_2(R), for n > 2 the equivalence between FP_n(R) and FH_n(R) is unknown. Consequently, it is not known whether type FH_n(R) is a quasi-isometry invariant property.

Establishing quasi-isometry invariance of FH_n(R) would clarify the relationship between algebraic finiteness properties and geometric actions on complexes and would support broader quasi-isometry invariance results for isoperimetric functions.

References

Another issue with assuming FH (Rn is that it is not known that if G and H are quasi-isometric, then G is of type FHn(R) if and only if H is of type FH nR). Thus we conclude with the following question. Question 2.23. Suppose G and H are quasi-isometric and G is of type FH nR). Is H of type FH (n)? It would suffice to show that FPn(R) is equivalent to FHn(R).

— Subgroups of word hyperbolic groups in dimension 2 over arbitrary rings  (2405.19866 - Bader et al., 2024) in Question 2.23, Section 2.2