Free groups are L²-subgroup rigid

In this paper, we introduce the notion of L²-subgroup rigid groups and demonstrate that free groups are L²-subgroup rigid. As a consequence, we establish the equivalence between compressibility, inertness, strong inertness, and L²-independence for a finitely generated subgroup of a free group, confirming a conjecture by Dicks and Ventura as well as the one by Antolin and Jaikin-Zapirain.