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Deformation theory and finite simple quotients of triangle groups I

Michael Larsen, Alexander Lubotzky, Claude Marion (2014)

Journal of the European Mathematical Society

Let 2 a b c with μ = 1 / a + 1 / b + 1 / c < 1 and let T = T a , b , c = x , y , z : x a = y b = z c = x y z = 1 be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of T ? (Classically, for ( a , b , c ) = ( 2 , 3 , 7 ) and more recently also for general ( a , b , c ) .) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially prove...

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