CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

Quadratic Integers and Coxeter Groups

  Published:1999-12-01
 Printed: Dec 1999
  • Norman W. Johnson
  • Asia Ivić Weiss
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   HTML   LaTeX   MathJax   PDF   PostScript  

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, 20F55, 20G20, 20H10, 22E40 show english descriptions Structure of modular groups and generalizations; arithmetic groups [See also 20H05, 20H10, 22E40]
Reflection and Coxeter groups [See also 22E40, 51F15]
Linear algebraic groups over the reals, the complexes, the quaternions
Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]
Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]
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]
 

© Canadian Mathematical Society, 2014 : http://www.cms.math.ca/