On Problem of Best Circle to Discontinuous Groups in Hyperbolic Plane

TL;DR

This paper investigates the largest inscribed circle in fundamental domains of certain discontinuous groups in the hyperbolic plane, providing methods to determine the maximum radius and geometric properties of such circles.

Contribution

It introduces a method using Lagrange multipliers to find the best inscribed circle for specific hyperbolic groups, extending to more general cases with multiple rotational centers.

Findings

Maximum radius achieved by equalizing angles at centers

Method applicable to groups with multiple rotational centers

Geometric characterization of inscribed circles in hyperbolic domains

Abstract

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.