On the commability and quasi-isometry classification of focal groups Yves Cornulier Primary 20F67Secondary 20E0820F6522D0522D4553C3057M0757S2057S30Compacting automorphismslocally compact groupsGromov-hyperbolic groupsfocal groupscommabilitymillefeuille spacesquasi-isometric classification We introduce the notion of commability between locally compact groups, namely the equivalence relation generated by co-compact inclusions and quotients by compact normal subgroups. We give a classification of focal hyperbolic locally compact groups up to commability. In the mixed case, it involves a real parameter, which is shown to be a quasi-isometry invariant. Indiana University Mathematics Journal 2015 text pdf 10.1512/iumj.2015.64.5441 10.1512/iumj.2015.64.5441 en Indiana Univ. Math. J. 64 (2015) 115 - 150 state-of-the-art mathematics http://iumj.org/access/