Computing fundamental domains for Fuchsian groups par

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 that: (i) ΓD = H, and (ii) gDo ∩Do = ∅ for all g ∈ Γ \ {1}, where o denotes the interior. Assume further that Γ has cofinite area, i.e., the coset space X = Γ\H has finite hyperbolic area μ(X) < ∞; then it follows that Γ is finitely generated. In this article, we exhibit an algorithm to compute a fundamental domain for Γ; we assume that Γ is specified by a finite set of generators G ⊂ SL2(K) with K →֒ R ∩Q a number field, and we call Γ exact. Suppose that p ∈ H has trivial stabilizer Γp = {1}. Then the set D(p) = {z ∈ H : d(z, p) ≤ d(gz, p) for all g ∈ Γ}, known as a Dirichlet domain, is a hyperbolically convex fundamental domain for Γ. The boundary of D(p) consists of finitely many geodesic segments or sides. We specify D(p) by a sequence of vertices, oriented counterclockwise around p. The domain D(p) has a natural side pairing : For each side s of D(p), there exists a unique side s∗ and g ∈ Γ \ {1} such that s∗ = gs, and the set of such g comprises a set of generators for Γ. Our main theorem is as follows. Theorem. There exists an algorithm which, given an exact Fuchsian group Γ with cofinite area and a point p ∈ H with Γp = {1}, returns the Dirichlet domain D(p), a side pairing for D(p), and a finite presentation for Γ with a minimal set of generators.