  Looking at the well understood case of log terminal surface singularities,
one observes that each of them is the quotient of a factorial one by a finite
solvable group. The derived series of this group reflects an iteration of Cox
rings of surface singularities. We extend this picture to log terminal
singularities in any dimension coming with a torus action of complexity one. In
this setting, the previously finite groups become solvable torus extensions. As
explicit examples, we investigate compound du Val threefold singularities. We
give a complete classification and exhibit all the possible chains of iterated
Cox rings.
