6533b86ffe1ef96bd12ccfee

RESEARCH PRODUCT

Finite quotients of the Picard group and related hyperbolic tetrahedral and Bianchi groups

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There is an extensive literature on the fi{}nite index subgroups and the fi{}nite quotient groups of the Picard group $PSL\left(2,\mathbb{Z}\mid i\mid\right)$. The main result of the present paper is the classifi{}cation of all linear fractional groups $PSL\left(2,p^{m}\right)$ which occur as fi{}nite quotients of the Picard group. We classify also the fi{}nite quotients of linear fractional type of various related hyperbolic tetrahedral groups which uniformize the cusped orientable hyperbolic 3-orbifolds of minimal volumes. Also these cusped tetrahedral groups are of Bianchi type, that is of the form $PSL\left(2,\mathbb{Z}\mid\omega\mid\right)$ or $PGL\left(2,\mathbb{Z}\mid\omega\mid\right)$, for suitable $\omega\epsilon\mathbb{C}.$ It turns out that all fi{}nite quotients of linear fractional type of these tetrahedral groups are obtained by reduction of matrix coeffi{}cients mod p whereas for the Picard group most quotients do not arise in this way (as in the case of the classical modular group $PSL\left(2,\mathbb{Z}\right)$. From a geometric point of view, we are looking for hyperbolic 3-manifolds which are regular coverings, with covering groups isomorphic to $PSL\left(2,q\right)$ or $PGL\left(2,q\right)$ and acting by isometries, of the cusped hyperbolic 3-orbifolds of minimal volumes. So these are the cusped hyperbolic 3-manifolds of minimal volumes admitting actions of linear fractional groups. We also give some application to the construction of closed hyperbolic 3-manifolds with large group actions. We are concentrating in this work on quotients of linear fractional type because all fi{}nite quotients of relatively small order of the above groups are of this or closely related types (similar to the case of Hurwitz actions on Riemann surfaces), so the linear fractional groups are the fi{}rst and most important class of fi{}nite simple groups to take into consideration.