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Painlevé equations and complex reflections

Philip Boalch (2003)

Annales de l’institut Fourier

We will explain how some new algebraic solutions of the sixth Painlevé equation arise from complex reflection groups, thereby extending some results of Hitchin and Dubrovin-- Mazzocco for real reflection groups. The problem of finding explicit formulae for these solutions will be addressed elsewhere.

Periodic subgroups of projective linear groups in positive characteristic

Alla Detinko, Dane Flannery (2008)

Open Mathematics

We classify the maximal irreducible periodic subgroups of PGL(q, 𝔽 ), where 𝔽 is a field of positive characteristic p transcendental over its prime subfield, q = p is prime, and 𝔽 × has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q, 𝔽 ) containing the centre 𝔽 ×1q of GL(q, 𝔽 ), such that G/ 𝔽 ×1q is a maximal periodic subgroup of PGL(q, 𝔽 ), and if H is another group of this kind then H is GL(q, 𝔽 )-conjugate to a group in the list. We give criteria for determining...

Principal congruence link complements

Mark D. Baker, Alan W. Reid (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

In this paper we study principal congruence link complements in S 3 . It is known that there are only finitely many such link complements, and we make a start on enumerating them using a combination of theoretical methods and computer calculations with MAGMA.

Proper periods of normal N.E.C. subgroups with even index.

Emilio Bujalance (1981)

Revista Matemática Hispanoamericana

By a non-Euclidean crystallographic (N.E.C.) group we shall mean a discrete subgroup Γ of isometries of the non-Euclidean plane including those reverse orientation, reflections and glide-reflections.In [1] we computed the proper periods of normal N.E.C. subgroups of an N.E.C. group, when the index of the group with respect to the subgroup is odd. In this paper we shall compute the proper period of normal N.E.C. subgroups, when the index is even.The corresponding problem for Fuchsian groups, which...

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