I will explain the terms in the title (and in this abstract), give some history, and sketch a new result. A century ago, fundamental results in dynamics and ergodic theory were achieved using a connection of regular continued fractions to the group SL(2,Z). Late in the previous century, proofs were given that Gauss's continued fraction map on the unit interval is a factor of the first return map of the geodesic flow to a 2-d set of unit tangent vectors on a surface related to SL(2,Z). With Calta and Kraaikamp, we showed similar results for infinitely many groups and interval maps. I have recently resolved a conjecture of ours, whose statement and proof I will briefly indicate.