We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group Γ with cofinite area. As a consequence, we compute the invariants of Γ, including an explicit finite presentation for Γ. Let Γ ⊂ PSL2(R) be a Fuchsian group, a discrete group of orientationpreserving isometries of the upper half-plane H with hyperbolic metric d. A fundamental domain for Γ is a closed domain D ⊂ H such th...

Let K be a totally real algebraic number field and O an order in a quaternion algebra A over K. Assume that the group O1 of units in O with reduced norm equal to 1 is embedded into PSL2(R) as an arithmetic Fuchsian group. It is shown how Ford’s algorithm can be effectively applied in order to determine a fundamental domain of O1 as well as a complete system of generators of O1.

In this paper, we verify the solvability of free product of finite cyclic groups with topological methods. We use Cayley graphs and Everitt methods to construct suitable 2-complexes corresponding to the presentations of groups and their commutator subgroups. In particular, using these methods, we prove that the commutator subgroup of $mathbb{Z}_{m}*mathbb{Z}_{n}$ is free of rank $(m-1)(n-1)$ fo...