Event
Claude Marion, University of Padova
Wednesday, May 10, 2017 10:45
Burnside Hall
Room 920, 805 rue Sherbrooke Ouest, Montreal, QC, H3A 0B9, CA
On finite simple images 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 Ta,b,c = hx, y, z : x a = y b = z c = xyz = 1i. 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 ja(G) = dim G[a] . Given a triple (a, b, c) of positive integers, we conjectured a few years ago that if ja(G) +jb(G) +jc(G) = 2 dim G 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.