Compact quantum metric spaces from quantum groups of rapid decay

We present a modified version of the definition of property RD for discrete quantum groups given by Vergnioux in order to accommodate examples of non-unimodular quantum groups. Moreover we extend the construction of spectral triples associated to discrete groups with length functions, originally due to Connes, to the setting of quantum groups. For quantum groups of rapid decay we study the resulting spectral triples from the point of view of compact quantum metric spaces in the sense of Rieffel.
Comment: 19 pages