The <HNN extension> is the <Baumslag-Solitar group>
$$
\operatorname{BS}(1,2)
=\langle u,t\mid tut^{-1}=u^2\rangle.
$$
Its <Bass-Serre tree> has vertices $G/\langle u\rangle$ and oriented edges $G/\langle u\rangle$, with the two endpoint maps induced by the identity embedding and the index-two embedding $n\mapsto2n$. At each vertex there is one incident edge on the identity side and two on the index-two side. The underlying unoriented tree is therefore infinite and $3$-regular, with an orientation in which every vertex has one incoming and two outgoing edges, up to reversing the convention.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 133 / 2 / a / ii / Solution

{c}
{wiki=HNN_extension}

Given isomorphic subgroups $C,D\leq A$ and an isomorphism $\phi:C\to D$, the HNN extension is
$$
\langle A,t\mid tct^{-1}=\phi(c)\text{ for every }c\in C\rangle.
$$
Its stable letter $t$ records the loop edge in the associated graph of groups.


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