Baumslag-Solitar group

Definition

Suppose $m,n$ are integers. The Baumslag-Solitar group $BS(m,n)$ is defined as a group with the following presentation:

$\!BS(m,n):=\langle a,b\mid ba^{m}b^{-1}=a^{n}\rangle$

Note that $BS(m,n)\cong BS(n,m)$ by identifying the $b$ of the first group with the $b^{-1}$ f the second. Also, $BS(m,n)\cong BS(-m,-n)$, so it suffices to consider pairs where at least one element is nonnegative.

Particular cases

$m$	$n$	What group do we get?
0	0	free group:F2
1	1	free abelian group of rank two
1	2	Baumslag-Solitar group:BS(1,2)
1	-1	fundamental group of Klein bottle