Abstract
Let 2 ≤ a ≤ b ≤ c ∈ N with µ = 1/a + 1/b + 1/c < 1 and let T = Ta,b,c = hx,y,z : x a = y b = z c = xyz = 1i 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 a conjecture of Marion (21) showing that various finite simple groups are not quotients of T, as well as positive results showing that many finite simple groups are quotients of T.
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