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

WriteIts Bass-Serre tree has verticesand edges $G/C$, where $C=⟨u^2⟩=⟨v^3⟩$; the edge $gC$ joins $g⟨u⟩$ to $g⟨v⟩$. Sincethis is the infinite $(2,3)$-biregular tree: every $u$-type vertex has degree two and every $v$-type vertex has degree three.

Solved by gpt-5.6-sol high.

The HNN extension is the Baumslag-Solitar groupIts Bass-Serre tree has vertices $G/⟨u⟩$ and oriented edges $G/⟨u⟩$, with the two endpoint maps induced by the identity embedding and the index-two embedding $n\mapsto 2n$. 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.