triangle groupsTriangleGroups2004-06-16 05:50:482006-07-06 11:12:22Definitionvon Dyck groups% this is the default PlanetMath preamble. as your knowledge
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% define commands hereConsider the following group presentation:
$$\Delta(l,m,n)=\langle a,b,c:a^2,b^2,c^2,(ab)^l,(bc)^n,(ca)^m\rangle$$
where $l,m,n\in\mathbb{N}$.
A group with this presentation corresponds to a triangle; roughly, the generators are reflections in its sides and its angles are $\pi/l,\pi/m,\pi/n$.
Denote by $D(l,m,n)$ the subgroup of \PMlinkname{index}{Coset} 2 in $\Delta(l,m,n)$, corresponding to preservation of \PMlinkescapetext{orientation} of the triangle.
The $D(l,m,n)$ are defined by the following presentation:
$$D(l,m,n)=\langle x,y:x^l,y^m,(xy)^n\rangle$$
Note that $D(l,m,n)\cong D(m,l,n)\cong D(n,m,l)$, so $D(l,m,n)$ is \PMlinkescapetext{independent of the order} of the $l,m,n$.
Arising from the geometrical nature of these groups, $$1/l+1/m+1/n>1$$is called the \emph{spherical case},$$1/l+1/m+1/n=1$$is called the \emph{Euclidean case}, and$$1/l+1/m+1/n<1$$is called the \emph{hyperbolic case}
Groups either of the form $\Delta(l,m,n)$ or $D(l,m,n)$ are referred to as \emph{triangle groups}; groups of the form $D(l,m,n)$ are sometimes refered to as \emph{von Dyck groups}.