Arithmetic Fuchsian groups of genus zero.
Let \(k\) be a totally real number field of degree \(n\), and let \(A\) be a quaternion algebra defined over \(k\) which is ramified at all infinite places except one. Let \(\rho\colon A\to M(2,\mathbb{R})\) be an embedding, \(G\) be an order of \(A\) and \(G^1\) the elements of norm one in \(G\). Then \(P\rho(G^1)<\text{PSL}(2,\mathbb{R})\) is a finite \(\omega\)-area Fuchsian group. A Fuchsian group \(\Gamma\) is defined to be arithmetic if and only if \(\Gamma\) is commensurable (in the wide sense) with some such \(P\rho(G^1)\). It is known that there are only finitely many conjugacy classes of arithmetic Fuchsian groups with a given signature or a given number of generators. On the other hand there are infinitely many non-conjugate subgroups of the modular group \(\text{PSL}(2,\mathbb{Z})\) of genus \(0\). Moreover, there are also infinitely many conjugacy classes of cocompact arithmetic Fuchsian groups of genus \(0\). Here the authors show, in contrast, that there are only finitely many conjugacy classes of maximal arithmetic Fuchsian groups of genus \(0\). They also construct genus \(0\) arithmetic Fuchsian groups arising from definining fields \(k\) of degrees \(\leq 7\) and conjecture that, if \(\Gamma\) is an arithmetic Fuchsian group of genus \(0\), then the degree of \(k\) over \(\mathbb{Q}\) is at most seven. It is known that the degree of \(k\) over \(\mathbb{Q}\) is \(\leq 10\).
scientific article; zbMATH DE number 5074495