Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 133 / 2 / a

Direct multiplication gives $A^3=B^2=-I$, so $⟨A⟩$ has order $6$, $⟨B⟩$ has order $4$, and their intersection is $\{I,-I\}$ of order $2$. For the amalgamated free productthe Bass-Serre tree has verticesand edges $S{L}_2(\mathbb{Z})/\{\pm I\}$. Each edge $g\{\pm I\}$ joins $g⟨A⟩$ to $g⟨B⟩$. Thus it is an infinite bipartite tree in which the $⟨A⟩$-vertices have degree $3$ and the $⟨B⟩$-vertices have degree $2$:with a third branch leaving every ${\circ }_3$ vertex and the same pattern continuing on every branch.