The Hanna Neumann conjecture for graphs of free groups with cyclic edge groups

Abstract The Hanna Neumann conjecture (HNC) for a free group G predicts that (U V) (U) (V) for all finitely generated subgroups U and V, where (H) = \- (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 L² -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 L² -Hall property for free and surface groups, which Brown and Kharlampovich extended to all limit groups. In this paper, we prove the L² -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 L² -Hall property for limit groups. As a corollary, we show that all these groups satisfy a strengthened version of the HNC.