Deformation theory and finite simple quotients of triangle groups I

Let 2 ≤ a ≤ b ≤ c ∈ ℕ with μ D 1/a + 1/b + 1/c < 1 and let T = T_a;b;c = 〈x, y, z: x_a = y_b = z_c = xyz = 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 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.