On Problem of Best Circle to Discontinuous Groups in Hyperbolic Plane

The aim of this paper is to describe the largest inscribed circle into the fundamental domains of a discontinuous group in Bolyai-Lobachevsky hyperbolic plane. We give some known basic facts related to the Poincare-Delone problem and the existence notion of the inscribed circle. We study the best circle of the group G = [3, 3, 3, 3] with 4 rotational centers each of order 3. Using the Lagrange multiplier method, we would describe the characteristic of the best-inscribed circle. The method could be applied for the more general case in G = [3, 3, 3,..., 3] with at least 4 rotational centers each of order 3, by more and more computations. We observed by a more geometric Theorem 2 that the maximum radius is attained by equalizing the angles at equivalent centers and the additional vertices with trivial stabilizers, respectively. Theorem 3 will close our arguments where Lemma 3 and 4 play key roles.