Congruence Subgroups in the Hurwitz Quaternion Order
نویسنده
چکیده
We clarify the explicit structure of the Hurwitz quaternion order, which is of fundamental importance in Riemann surface theory and systolic geometry. We present some properties of the associated congruence subgroups. Namely, we show that a Hurwitz group defined by a congruence subgroup associated with an odd ideal, is necessarily defined by a principal congruence subgroup. All such Hurwitz groups have the form PSL2(L), for a suitable semilocal ring L. A generalisation for congruence towers of arithmetic Riemann surfaces is presented.
منابع مشابه
Bolza Quaternion Order and Asymptotics of Systoles Along Congruence Subgroups
We give a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order. This description enables us to show that the corresponding principal congruence covers satisfy the bound sys(X) > 4 3 log g(X) on the systole, where g is the genus. We also exhibit the Bolza group as a congruence subgroup, and calculate out a few examples of “Bolza twins” (u...
متن کاملLogarithmic Growth of Systole of Arithmetic Riemann Surfaces along Congruence Subgroups
P. Buser and P. Sarnak constructed Riemann surfaces whose systole behaves logarithmically in the genus. The Fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field. We generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces. The key tool is a new trace estimate valid for an arbitrary idea...
متن کاملCongruence subgroups of the minimal covolume arithmetic Kleinian group
We identify the normal subgroups of the orientation preserving subgroup [3, 5, 3] of the Coxeter group [3, 5, 3], with the factor group isomorphic to PSL2(Fq) with particular congruence subgroups of an arithmetic subgroup of PSL2(C) derived from a quaternion algebra over a quartic field. 1 Motivation – Hurwitz groups and HurwitzMacbeath surfaces It is a well known fact that up to isomorphy, the...
متن کاملPresentations for Quaternionic S-Unit Groups
In this paper, we give an algorithm for presenting S-unit groups of an order O in a definite rational quaternion algebra B such that, for every p ∈ S at which B splits, the localization of O at p is maximal and all left ideals of O of norm p are principal. We then apply this to give presentations for projective S-unit groups of the Hurwitz order in Hamilton’s quaternions over the rational field...
متن کاملFuzzy subgroups of the direct product of a generalized quaternion group and a cyclic group of any odd order
Bentea and Tu{a}rnu{a}uceanu~(An. c{S}tiinc{t}. Univ. Al. I.Cuza Iac{s}, Ser. Nouv{a}, Mat., {bf 54(1)} (2008), 209-220)proposed the following problem: Find an explicit formula for thenumber of fuzzy subgroups of a finite hamiltonian group of type$Q_8times mathbb{Z}_n$ where $Q_8$ is the quaternion group oforder $8$ and $n$ is an arbitrary odd integer. In this paper weconsider more general grou...
متن کامل