Relative ($τ$), Expanders, and Decay of Correlations for certain Expanding Maps

Relative ($τ$) is equivalent to a statement that the sequence of Cayley graphs associated to group quotients $Γ_q=Γ/N_q$, $q\in\mathbb{N}$, form an expander family. There is a philosophy that expander graphs give rise to good mixing; for instance, one has exponential mixing for the geodesic flow uni