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# Quadratic Integers and Coxeter Groups

Published:1999-12-01
Printed: Dec 1999
• Norman W. Johnson
• Asia Ivić Weiss
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## Abstract

Matrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive $n$-space or hyperbolic $(n+1)$-space $\mbox{H}^{n+1}$. For small $n$, these may be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of $\mbox{H}^{n+1}$. We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group $\PSL_2 (\bbZ)$, the Gaussian modular (Picard) group $\PSL_2 (\bbZ[{\it i}])$, and the Eisenstein modular group $\PSL_2 (\bbZ[\omega ])$.
 MSC Classifications: 11F06 - Structure of modular groups and generalizations; arithmetic groups [See also 20H05, 20H10, 22E40] 20F55 - Reflection and Coxeter groups [See also 22E40, 51F15] 20G20 - Linear algebraic groups over the reals, the complexes, the quaternions 20H10 - Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx] 22E40 - Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]

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