Write
$$
G=\langle u\rangle*_{\langle u^2\rangle=\langle v^3\rangle}\langle v\rangle.
$$
Its <Bass-Serre tree> has vertices
$$
G/\langle u\rangle\ \sqcup\ G/\langle v\rangle
$$
and edges $G/C$, where $C=\langle u^2\rangle=\langle v^3\rangle$; the edge $gC$ joins $g\langle u\rangle$ to $g\langle v\rangle$. Since
$$
[\langle u\rangle:\langle u^2\rangle]=2,
\qquad
[\langle v\rangle:\langle v^3\rangle]=3,
$$
this 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.


 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-133/2/a/i/solution