I am reading an article of Baumslag:
Baumslag, Gilbert, "Finitely generated cyclic extensions of free groups are residually finite." Bull. Austral. Math. Soc. 5 (1971), 87–94.
and he mentions that many one-relator groups, in particular, fundamental groups of surfaces, are free-by-cyclic, see picture. Could somebody comment on this: how are surface groups free-by-cyclic? Or the one-relator groups mentioned by Baumslag?
The orientable surface group is free-by-cyclic: if the standard generators are $x_1,...,x_g, y_1,...,y_g$ then the homomorphism onto the cyclic group $\langle x_1\rangle$ which kills all other generators is onto and its kernel is of infinite index, whence free.
The group $\langle a,b,c| c^n=[a,b]\rangle =\langle a,b,c| b^a=c^nb\rangle$ and many other $HNN$-extensions of a free group with cyclic associated subgroups is also free-by-cyclic. It has a homomorphism onto the cyclic group generated by the free letter whose kernel is free. The proof can be found here.