Function germs defined on isolated hypersurface singularities
We describe a process for defining and computing a fundamental domain in the upper half plane of a Shimura curve associated with an order in a quaternion algebra . A fundamental domain for realizes a finite presentation of the quaternion unit group, modulo units of its center. We give explicit examples of domains for the curves . The first example is a classical example of a triangle group and the second is a corrected version of that appearing in the book of Vignéras [13], due to Michon....
This is a brief exposition on the uses of non-commutative fundamental groups in the study of Diophantine problems.
Continuing our work on the fundamental groups of conic-line arrangements (Amram et al., 2003), we obtain presentations of fundamental groups of the complements of three families of quadric arrangements in P2. The first arrangement is a union of n conics, which are tangent to each other at two common points. The second arrangement is composed of n quadrics which are tangent to each other at one common point. The third arrangement is composed of n quadrics, n-1 of them are tangent to the n-th one...