# 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:

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

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