Abstract
A polyhedral group G is defined to be the orientation-preserving subgroup of a discrete reflection group acting on hyperbolic 3-space H 3, and having a fundamental polyhedron of finite volume. A special presentation for G is obtained from the geometry of the polyhedron. This gives G the structure of a graph amalgamation product, and which, in some cases, splits as a free product with amalgamation. The simplest examples of polyhedral groups are the so-called tetrahedral groups. Other examples are given amongst the the groups PGL(2, O m ), where O m is the ring of algebraic integers in the quadratic imaginary field Q( -m ) , m>0.
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