Torsion-freeness of the group $\langle a, b \mid a b^m = ba^n\rangle$ For integers $m$ and $n$ let $K(m,n)$ be the group $\langle a, b \mid a b^m = ba^n\rangle$. Is there a special name for this group? Is there a complete characterization of those pairs $(m,n)$ for which the group is torsion-free? Or bi-orderable (in the sense of Rolfsen: http://www.math.ubc.ca/~rolfsen/papers/luminynotes/lum.pdf)?
 A: These groups are torsion-free groups with a single defining relation, which implies that they are right-orderable. I do not know if they are bi-orderable though.
A one-relator group is a group with a presentation of the form $\langle X; S\rangle$ where $X$ is some set (possibly infinite) and $S$ is an arbitrary word. Your groups are clearly one-relator groups, and they are torsion-free because every one-relator group has torsion if and only if the relator $S$ is a proper power of some other element $R$ in $F(X)$, that is, $S\equiv R^n$ for $n>1$. This follows from the fact that you can always use HNN-extensions and free products with amalgamation to write such a group in terms of another one-relator group where the relator has strictly shorter length. This gives you an induction step. This induction idea is called Magnus' method, after Wilhelm Magnus, and Lyndon and Schupp's book goes into this interpretation of Magnus' method a wee bit, but they prefer Magnus' original (rather messy) approach. If you want a neat proof, look up the paper The Freiheitsstatz and its extensions of Fine and Rosenberger. This is in a book though, so you could look up the paper of McCool and Schupp entitled One-relator groups and HNN-extensions.
These groups are locally-indicable because they are torsion-free one-relator groups (see the paper On locally indicable groups, 1982, by Jim Howie), that is, every finitely-generated subgroup maps into the infinite cyclic group. This means that they are right-orderable (see the paper A note on group rings of certain torsion free groups, 1972, by Burns and Hale). I do not know if they are bi-orderable though.
EDIT: I have just looked over the notes you link too, and you should realise that my last paragraph is expanded on in the notes.
A: Proposition 5.18 from Lyndon, Schupp, Combinatorial Group Theory,  Springer, 1977:
If $G = \langle X; r=1\rangle$ and $r$ is not a proper power, then $G$ is torsion free. 
