Controlling the Galois images in one-dimensional families of ℓ-adic representations

Let k be a finitely generated field of characteristic 0 and ℓ a prime. Let S be a smooth, separated and geometrically connected scheme over k and let ρ:π1(S)→GLr(Zℓ) be an ℓ-adic representation of the étale fundamental group of S. Let G and G¯ denote the images of π1(S) and π1(Sk¯) respectively. Given a closed point s∈S, we write Gs for the image of the absolute Galois group Γk(s) of the residue field k(s) viewed as a decomposition group at s in π1(S). By previous works of the authors, it is known that, when S is a curve, for all d≥1 and all but finitely many s∈S with [k(s):k]≤d, the codimension of Gs in G is at most 2. In this note, we improve this rigidity result as follows. Write g, g¯, gs for the Lie algebras of G, G¯, Gs respectively. Then for all but finitely many s∈S with [k(s):k]≤d, one of the following holds: (i) the codimension of Gs in G is at most 1 and gs contains D(g¯):=[g¯,g¯]; or (ii) the codimension of Gs in G is 2 and gs contains D2(g¯):=D(D(g¯)). We also obtain an arithmetic variant of this result, which involves the derived series of g.