The Hanna Neumann conjecture (HNC) for a free group G predicts that $\overline{\chi}(U\cap V)\leqslant \overline{\chi} (U)\overline{\chi}(V)$ for all finitely generated subgroups U and V, where $\overline{\chi}(H) = \max\{-\chi(H),0\}$ denotes the reduced Euler characteristic of H. A strengthened version of the HNC was proved independently by Friedman and Mineyev in 2011. Recently, Antolín and Jaikin-Zapirain introduced the $L^2$-Hall property and showed that if G is a hyperbolic limit group that satisfies this property, then G satisfies the HNC. Antolín and Jaikin-Zapirain established the $L^2$-Hall property for free and surface groups, which Brown and Kharlampovich extended to all limit groups. In this paper, we prove the $L^2$-Hall property for graphs of free groups with cyclic edge groups that are hyperbolic relative to virtually abelian subgroups. We also give another proof of the $L^2$-Hall property for limit groups. As a corollary, we show that all these groups satisfy a strengthened version of the HNC.

The Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957. In 2011, a strengthened version of the conjecture was proved independently by Joel Friedman and by Igor Mineyev. In this paper we show that the strengthened Hanna Neumann conjecture holds not only in free groups but also in non-solvable surface groups. In addition, we show that a retract in a free group and in a surface group is inert. This implies the Dicks–Ventura inertia conjecture for free and surface groups.