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  "body": "\n## Example\n\nLet $m$ and $n$ be nonzero integers.\nThe *Baumslag–Solitar group* $BS(m, n)$ is the group with the [[presentation|0JGEQXQ4AC]]\n$$BS(m, n) = \\langle a, b \\mid b a^m b^{-1} = a^n \\rangle.$$\nIt is a [[one-relator group|68AFPHR4GN]], since the presentation has the single relator $b a^m b^{-1} a^{-n}$.\nIt is also the [[HNN extension|YEYERRY37K]] of the infinite cyclic group $\\langle a \\rangle$ that conjugates $\\langle a^m \\rangle$ onto $\\langle a^n \\rangle$ by $a^m \\mapsto a^n$, with stable letter $b$.\n\n## Remarks\n\nSmall cases: $BS(1, 1) \\cong \\Z^2$, and $BS(1, -1)$ is the fundamental group of the Klein bottle.\nThe group $BS(1, n)$ is isomorphic to the semidirect product $\\Z[1/n] \\rtimes \\Z$, where the generator of $\\Z$ acts by multiplication by $n$.\nThe isomorphism sends $a$ to $1 \\in \\Z[1/n]$ and $b$ to the generator of $\\Z$.\nIn particular $BS(1, n)$ is solvable.\n\n$BS(m, n)$ is [[residually finite|HEE922EHQ8]] if and only if $\\abs{m} = 1$, $\\abs{n} = 1$ or $\\abs{m} = \\abs{n}$.\n\n$BS(2, 3) = \\langle a, b \\mid b a^2 b^{-1} = a^3 \\rangle$ is not [[Hopfian|PHEN4E1JPT]].\nLet $\\varphi$ be the endomorphism with $\\varphi(a) = a^2$ and $\\varphi(b) = b$.\nIt is well defined, since $\\varphi(b a^2 b^{-1}) = b a^4 b^{-1} = (b a^2 b^{-1})^2 = a^6 = \\varphi(a^3)$.\nIt is surjective, since $a = a^3 a^{-2} = b a^2 b^{-1} a^{-2} = \\varphi(b a b^{-1} a^{-1})$.\nIt is not injective: the element $[b a b^{-1}, a]$ is not $1$ by Britton's lemma, but $\\varphi$ sends it to $[b a^2 b^{-1}, a^2] = [a^3, a^2] = 1$.\n\n## Explanations\n\n### L1 human\n\nA Baumslag–Solitar group has two generators and one rule: conjugating $a^m$ by $b$ gives $a^n$.\nFor $m = n = 1$ the rule says that $a$ and $b$ commute, and the group is $\\Z^2$.\nFor $m = 1$ and $n = 2$, think of $a$ as the shift $x \\mapsto x + 1$ and $b$ as the doubling $x \\mapsto 2x$ of the real line.\nThen $b a b^{-1}$ is $x \\mapsto x + 2$, which is $a^2$.\nOther values of $m$ and $n$ give groups with surprising properties, and they serve as standard test cases in group theory.\n",
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