Free groups are $L^{2}$-subgroup rigid

In this paper, we introduce the notion of L^{2} -subgroup rigid groups and demonstrate that free groups are L^{2} -subgroup rigid. As a consequence, we establish the equivalence between compressibility, inertness, strong inertness, and L^{2} -independence for a finitely generated subgroup of a free group, confirming a conjecture by Dicks and Ventura as well as the one by Antolín and Jaikin-Zapirain.