Let $B$ be a group and let $A_1$ and $A_2$ subgroups of $B$ with $\phi :A_1\rightarrow A_2$ an isomorphism. Let $\left<t\right>$ be the infinite cyclic group, generated by a new element $t$. The $\mathbf{ HNN}$- extension $H$ is formed by adjoined to the free product $B* \left<t\right>$ the relations $ta t^{-1}=\phi(a)$ for all $a\in A_1$. Hence, $H$ has presentation $$H=\left<B,t\mid ta t^{-1}=\phi(a), \ a\in A_1\right>.$$ The group $B$ is called the base of $H$, $t$ is called the stable latter, and $A_1$ and $A_2$ is called the associated groups. An $\mathbf{HNN}$-extension $H$ is called an ascending $\mathbf{HNN}$-extension, if at least one of the subgroup $A_1$ and $A_2$ is equal to the base $B$. Bieri and Strebel showed that if $N$ is a normal subgroup of a finitely presented solvable group $G$ and $G/N$ is infinite cyclic, then $G$ is an ascending $\mathbf{ HNN}$-extension with a finitely generated base $B$.
My question is: if we have an ascending $\mathbf{ HNN}$-extension $H$ with a f.g. base $B$, then When $H$ is finitely presented group?
