Dimension invariants of outer automorphism groups

The geometric dimension for proper actions (gd) under bar (G) of a group G is the minimal dimension of a classifying space for proper actions (gd) under barG. We construct for every integer r >= 1, an example of a virtually torsion- free Gromov-hyperbolicgroup G such that for every group. which contains G as a finite index normal sub group, the virtual cohomological dimension vcd(Gamma) of Gamma equals (gd) under bar(Gamma) but such that the outer automorphism group Out(G) is virtually torsion- free, admits a cocompact model for (gd) under bar Out(G) but nonetheless has vcd(Out)(G)) <= gd(Out)(G)) - r