Baumslag-Solitar group

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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