  We show that if $\{\rho_{\ell}\}$ is a compatible system of absolutely
irreducible Galois representations of a number field then the residual
representation $\overline{\rho}_{\ell}$ is absolutely irreducible for $\ell$ in
a density 1 set of primes. The key technical result is the following theorem:
the image of $\rho_{\ell}$ is an open subgroup of a hyperspecial maximal
compact subgroup of its Zariski closure with bounded index (as $\ell$ varies).
This result combines a theorem of Larsen on the semi-simple part of the image
with an analogous result for the central torus that was recently proved by
Barnet-Lamb, Gee, Geraghty, and Taylor, and for which we give a new proof.
