Corrales Rodrigáñez, CarmenJespers, EricLeal, GuilhermeRio, Ángel del2023-06-202023-06-202004[1] A.F. Beardon, The Geometry of Discrete Groups, springer, Berlin, 1983. [2] L. Bianchi, Sui gruppi de sostituzioni lineari con coeficienti appartenenti a corpi quadratici imaginari, Math. Ann. 40 (1892) 332–412. [3] J. Elstrodt, F. Grunewald, J. Mennicke, Groups Acting on Hyperbolic Space, Harmonic Analysis and Number Theory, Springer, Berlin, 1998. [4] B. Fein, B. Gordon, J.M. Smith, On the representation of 1 as a sum of two squares in an algebraic number field, J. Number Theory 3 (1971) 310–315. [5] B. Fine, The Algebraic Theory of the Bianchi Groups, Marcel Dekker, New York, 1989. [6] A.J. Hahn, O.T. O’Meara, The Classical Groups and K-Theory, Grundlehren der mathematischen Wissenschaften 291, Springer, Heidelberg, 1989. [7] E. Jespers, Units in integral group rings: a survey, Proceedings of the International Conference on Methods in Ring Theory, Trento, 1997. Lecture Notes in Pure and Applied Mathematics, Vol. 198,Marcel Dekker, New York, 1998, pp. 141–169. [8] E. Kleinert, Units in Skew Fields, Progress in Mathematics, 186, Birkha¨ user Verlag, Basel, 2000. [9] E. Kleinert, Units of classical orders: a survey, Enseign. Math. (2) 40 (3–4) (1994) 205–248. [10] H. Poincare´, Me´moire sur les groupes kleine´ es, Acta. Math. 3 (1883) 49–92. [11] R. Riley, Applications of a computer implementation of Poincare´ ’s theorem on fundamental polyhedra, Math. Comp. 40 (162) (1983) 607–632.0001-870810.1016/j.aim.2003.07.015https://hdl.handle.net/20.500.14352/49857We give an algorithm to determine a finite set of generators of the unit group of an order in a non-split classical quaternion algebra H(K) over an imaginary quadratic extension K of the rationals. We then apply this method to obtain a presentation for the unit group of H(Z[(1+root-7)/(2)]). As a consequence a presentation is discovered for the orthogonal group SO3(Z[(1+root-7)/(2)]). These results provide the first examples of a characterization of the unit group of some group rings that have an epimorphic image that is an order in a non-commutative division algebra that is not a totally definite quaternion algebra.engjournal articlehttp://www.sciencedirect.com/science/article/pii/S0001870803002585http://www.sciencedirect.com/restricted access512.54AlgorithmsFinite sets of generatorsUnit groupsOrdersquaternion algebrasPresentationsGrupos (Matemáticas)