First-commutator compact quantum metrics for rapid-decay groups

Determine whether one may always choose a single commutator, that is, k=1, in the iterated-commutator construction of the compact quantum metric space for every finitely generated pair (G, ell) with rapid decay.

Background

For a proper length function ell on a countable discrete group G, the operator D defined by D delta_g = ell(g) delta_g yields the first-commutator seminorm L(f) = ||[D, lambda(f)]|| on C_c(G). The compact quantum metric property asks whether this seminorm induces the weak-* topology on the state space.

The paper explains that for pairs (G, ell) with rapid decay, existing results guarantee a compact quantum metric space after replacing the first commutator by a suitable fixed number k of iterated commutators. The unresolved issue is whether this higher-order construction is unnecessary for finitely generated rapid-decay pairs, so that k=1 always suffices. A later counterexample for a finitely generated group with a word length function lies outside the rapid-decay setting and therefore does not resolve this question.