19.3 (2018)

Let $G = A \ast_H B$ be the generalized free product of groups $A$ and $B$ with amalgamated subgroup $H$. Is $\psi(G) = 1$ in the following cases?
$\qquad$ a) $H$ is finite cyclic and $H_G = 1$.
$\qquad$ b) $H$ is finite cyclic and either $\lambda(A) \cap H_G = 1$ or $\lambda(B) \cap H_G = 1$.
$\qquad$ c) $H_G = 1$ and $H$ satisfies the minimum condition on subgroups.
$\qquad$ d) $\lambda(G) \cap H = 1$.
$\qquad$ e) $A$ and $B$ are free groups, $H$ is finitely generated and at least one of $|A : H|$ or $|B : H|$ is infinite.
$\qquad$ f) $H$ is infinite cyclic and is a retract of $A$ and $B$.
$\qquad$ g) $G$ nearly splits over $H$, and $H$ is a normal subgroup of $G$ of prime order.