Abstract: According to a famous theorem of Ornstein and Weiss, all ergodic pmp actions of countable amenable groups are orbit equivalent. This implies in particular that orbit equivalence forgets about all geometric properties of the groups. However some geometric information can be restored by imposing integrability conditions on the cocycles. The first instance of this phenomenon is a theorem of Bowen saying volume growth is preserved by L1-OE. We prove that this holds for the Folner function as well. On the other direction, we introduce new ways to construct OE couplings between amenable, or even sofic groups with prescribed integrability conditions. This is joint work with Delabie, Koivisto, Le Maître and Carderi.