Triangle groups as subgroups of unitary groups (Q5958876)

The triangle groups \(T(2,3,k)=\langle x,y\mid x^2=y^3=(xy)^k=1\rangle\) are shown to be subgroups of a unitary group \(\text{PSU}(2,\mathbb{Z}[\varepsilon])\) defined by a suitable Hermitian form, where the complex root \(\varepsilon\) of 1 has order \(k\) for odd \(k\) and order \(2k\) for even \(k\). Th authors prove the following main result: \(T(2,3,k)\) coincides with \(\text{PSU}(2,\mathbb{Z}[\varepsilon])\) precisely if \(k\in\{2,3,4,5,7,9,11\}\).NEWLINENEWLINENEWLINEIn the proof, \(\text{SU}(2,\mathbb{Z}[\varepsilon])\) is identified with the norm-1-group of a subring of a quaternion algebra over \(\mathbb{Q}(\varepsilon+\varepsilon^{-1})\). For \(k\leq 5\), the groups are finite, and the proof is achieved by comparing orders. The case \(k=6\) is special, here the Hermitian form is degenerate. For \(k=7,9\), or 11, the proof depends on very precise computations in order to find all squares in \(\mathbb{Z}[\varepsilon+\varepsilon^{-1}]\) with small traces. For the other values of \(k\), the authors show that \(T(2,3,k)\) does not contain a copy of \(\mathbb{Z}^2\), whereas \(\text{SU}(2,\mathbb{Z}[\varepsilon])\) does.