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Search: MSC category 20H10 ( Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx] )

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1. CJM 1999 (vol 51 pp. 1307)

Johnson, Norman W.; Weiss, Asia Ivić
Quadratic Integers and Coxeter Groups
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 $\PSL_2 (\bbZ[\omega ])$.

Categories:11F06, 20F55, 20G20, 20H10, 22E40

2. CJM 1997 (vol 49 pp. 405)

Vulakh, L. Ya.
On Hurwitz constants for Fuchsian groups
Explicit bounds for the Hurwitz constants for general cofinite Fuchsian groups have been found. It is shown that the bounds obtained are exact for the Hecke groups and triangular groups with signature $(0:2,p,q)$.

Categories:11J04, 20H10

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