Claude Marion "On finite simple quotients of triangle groups"

Given a triple (a, b, c) of positive integers, a finite group is said to be an (a, b, c)-group if it is a quotient of the triangle group
T_a, b, c = ⟨x, y, z : x^a = y^b = z^c = xyz = 1⟩.
Let G₀ = G(p^r) be a finite quasisimple group of Lie type with corresponding simple algebraic group G. Given a positive integer a, let G_[a] = {g ∈ G : g^a = 1} be the subvariety of G consisting of elements of order dividing a, and set j_a(G)=dimG_[a]. Given a triple (a, b, c) of positive integers, we conjectured a few years ago that if j_a(G)+j_b(G)+j_c(G)=2dimG then given a prime p there are only finitely many positive integers r such that G(p^r) is an (a, b, c)-group. We present some recent progress on this conjecture and related results: in particular the conjecture holds for finite simple groups.