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?



When is an HNN extension a free group?
Yes. This is a theorem of [Shenitzer](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=RT&pg6=RT&pg7=ALLF&pg8=ET&review_format=html&s4=shenitzer&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=13&mx-pid=69174). For a modern treatment see, for instance, this [recent paper of Louder](http://arxiv.org/abs/math/0612222). I give a proof of a similar fact in section 2 of [this preprint](http://arxiv.org/abs/1102.2866).


