Fundamental polyhedra for Fuchsian groups of signature \((0,4;p,2,2,2)\) (Q2763650)

This is a carefully and nicely written PhD-dissertation on the explicit construction of fundamental polyhedra of Fuchsian groups \(G\) (discrete subgroups of the orientation preserving isometry group \(I^+\) of the hyperbolic plane), for their actions by left multiplication on the isometry group \(I^+ \cong {\text{PSL}}_2(\mathbb R)\) (so the quotient spaces are certain 3-dimensional Seifert fiber spaces belonging to Thurston's \(\widetilde {\text{SL}}_2(\mathbb R)\)-geometry). From such a fundamental polyhedron, the quotient space \(G \setminus I^+\) can be constructed by identification, induced by the action of \(G\), of the faces of its boundary. In a previous dissertation, also from the group of Brieskorn at Bonn, Thomas Fischer has introduced a construction of such a fundamental polyhedron for the case that \(G\) contains elliptic elements, by considering the preimage \(\tilde G\) of \(G\) in the 2-fold covering of \(I^+ \cong {\text{PSU}}(1,1)\) (the isometry group of the Poincaré unit disk model) by the 3-dimensional unit pseudo-3-sphere \(\mathbb S \cong {\text{SU}}(1,1)\) associated to the standard bilinear form of signature (2,2) on \(\mathbb R^4\), and extending linearly the group structure of \(\mathbb S\) to an \(\mathbb R\)-algebra structure on \({\mathbb R}^4\) (a model case is that of the actions of the binary polyhedral groups on the 3-sphere \(S^3 \cong \text{SU}(2)\)). Motivated also by hypersurface singularities associated to such groups, in the present work fundamental polyhedra and their boundary identifications are determined for Fuchsian groups of signatures \((0;p,2,2,2)\), for \(p=3,4\) and \(5\) (i.e. the quotients of the hyperbolic plane by these groups have genus zero and four branch points of orders \(p\), 2,2 and 2). In previous work of the Bonn-group, the case of certain triangle groups \((0;p_1,p_2,p_3)\) has been considered. A main difference to these cases is that the groups of signature \((0;p,2,2,2)\) have a non-trivial 2-dimensional moduli space, and Fricke coordinates are used in the present work for the parametrization of these groups. For a fixed signature, the combinatorial types of the polyhedra depend on the specific group; there are finitely many combinatorial types for \(p=3\) and 4, and infinite countably many types for \(p=4\). NEWLINENEWLINENEWLINEAs noted above, a motivation comes from the theory of singularities. Roughly, a group \(G\) acts on suitable complex line bundles over the Poincaré disk; contracting the 0-section in the quotient \(G \setminus L\) one obtains a complex space with an isolated singular point 0. In two papers, Dolgachev considered the case of hypersurface singularities obtaining exactly 14 triangle groups \((0;p_1,p_2,p_3)\) giving the 14 quasihomogeneous unimodular exceptional singularities, and six signatures of type \((0;p_1,p_2,p_3,p_4)\) (including the three cases considered in the present paper) giving the six 1-parameter families of quasihomogeneous bimodular singularities in the Arnold-classification.