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

Abstract : There is an extensive literature on the finite index subgroups and the finite quotient groups of the Picard group $PSL(2,{\mathbb Z}[i])$. The main result of the present paper is the classification of all linear fractional groups $PSL(2,p^m)$ which occur as finite quotients of the Picard group. We classify also the finite 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 $PSL(2,{\mathbb Z}[\omega])$ or $PGL(2,{\mathbb Z}[\omega])$, for suitable $\omega\in{\mathbb C}$. It turns out that all finite quotients of linear fractional type of these tetrahedral groups are obtained by reduction of matrix coefficients mod $p$ whereas for the Picard group most quotients do not arise in this way (as in the case of the classical modular group $PSL(2,{\mathbb Z})$). From a geometric point of view, we are looking for hyperbolic 3-manifolds which are regular coverings, with covering groups isomorphic to $PSL(2,q)$ or $PGL(2,q)$ 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 finite 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 first and most important class of finite simple groups to take into consideration.
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https://hal.archives-ouvertes.fr/hal-00638126
Contributor : Luisa Paoluzzi <>
Submitted on : Friday, November 4, 2011 - 12:06:01 AM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM

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Luisa Paoluzzi, Bruno Zimmermann. Finite quotients of the Picard group and related hyperbolic tetrahedral and Bianchi groups. Rend. Istit. Mat. Univ. Trieste., 2001, 32, pp.257-288. ⟨hal-00638126⟩

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