TITLE: When is an HNN extension a free group?
QUESTION [6 upvotes]: Let $A$ be a free group and $G = A*_t$. When is $G$ also a free group? 
Suppose $t y t^{-1} = z$ and there is a splitting $A = B*C$ so that $y \in B$ and $z \in C$ and $z$ is a member of some basis of $C$ then clearly $G$ is free. Is this the only case that  $G$ is free?

REPLY [10 votes]: Yes.  This is a theorem of Shenitzer.  For a modern treatment see, for instance, this recent paper of Louder.  I give a proof of a similar fact in section 2 of this preprint.