Does the following group have a name? Is it amenable?


Fix $p$ and $q$



> 
> $\langle g,h: hg^qh^{-1}=gh^pg^{-1}\rangle$
> 
> 
> 



Certain finitely presented group
For 1-related groups, over a 2-letter alphabet, draw the relator on the plane grid: $g$'s are horizontal, $h$'s are vertical, starting at the point $O=(0,0)$. Connect $O$ with the endpoint $М$ of the resulting path $P$ by a vector $\vec{v}=\vec{OM}$. Consider the two support lines of the path $P$ that are parallel to $\vec{v}$. If both support lines intersect the path only once, the group is free-by-cyclic. If only one of the support lines intersects the path once, the group is an ascending HNN extension of a free group, and the rank of the free group is easily computed using Magnus rewriting. It can be all found in the old paper Kenneth S. Brown, Trees, valuations, and the Bieri-Neumann-Strebel invariant. Invent. Math., 90(3):479--504, 1987. 


