Quasi-isometry invariance of the first-commutator compact quantum metric property

Establish whether the compact quantum metric property of the first-commutator seminorm L(f) = ||[D, lambda(f)]|| on C_c(G) is invariant under quasi-isometric changes of the length function on a group.

Background

The main theorem proves that a proper length function whose induced left-invariant metric is hyperbolic and weakly geodesic produces a compact quantum metric space through the first commutator with the length operator. Proper weakly geodesic length functions on hyperbolic groups are quasi-isometric to word length functions, and the resulting length functions also have rapid decay.

However, rapid decay alone does not transfer the compact quantum metric conclusion from a word length to a quasi-isometric length. The paper explicitly identifies the missing issue as invariance of the compact quantum metric property for the first commutator under quasi-isometric changes of length function.