###### Options

# Finite Quotients of the Picard Group and Related Hyperbolic Tetrahedral and Bianchi Groups

Zimmermann, Bruno

Paoluzzi, Luisa

2001

Abstract

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.

Series

Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics

32 suppl. 1 (2001)

Publisher

Università degli Studi di Trieste. Dipartimento di Scienze Matematiche

Source

Luisa Paoluzzi and Bruno Zimmermann, "Finite quotients of the Picard group and related hyperbolic tetrahedral and Bianchi groups", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 32 (2001) suppl.1, pp. 257–288.

Languages

en

File(s)