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On some generalized triangle groups and three-dimensional orbifolds. (English) Zbl 0867.22009

Summary: Generalized triangle groups of the form \[ \Gamma(k,l;m) =\bigl\langle x,y \mid x^k= y^l= (xyx^{-1} yxy^{-1})^m =e\bigr\rangle \quad (2\leq k,l,m \leq\infty) \] are studied. For \(1/k+1/l+1/m>1\) the group \(\Gamma(k,l;m)\) is the fundamental group of a three-dimensional orbifold, whose underlying topological space is a sphere and the graph of singularities is one of the simplest connected graphs with two vertices. All the triples \((k,l;m)\) for which \(\Gamma (k,l; m)\) is finite are found. It is proved that, if at most one of the numbers \(k,l,m\) is equal to 2 and \((k,l;m) \neq(2,3;3)\) up to a permutation of \(k\) and \(l\), then \(\Gamma (k,l;m)\) is realized as a discrete group of motions of Lobachevsky space, so that for \(1/k+1/l+1/m>1\) the quotient space is equal to the above mentioned orbifold.

MSC:

22E40 Discrete subgroups of Lie groups
20H15 Other geometric groups, including crystallographic groups
57M05 Fundamental group, presentations, free differential calculus
57M50 General geometric structures on low-dimensional manifolds
57M60 Group actions on manifolds and cell complexes in low dimensions
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