  The fractal dimension of domain walls produced by changing the boundary
conditions from periodic to anti-periodic in one spatial direction is studied
using both the strong-disorder renormalization group and the greedy algorithm
for the Edwards-Anderson Ising spin-glass model for up to six space dimensions.
We find that for five or less space dimensions, the fractal dimension is less
than the space dimension. This means that interfaces are not space filling,
thus implying replica symmetry breaking is absent in space dimensions fewer
than six. However, the fractal dimension approaches the space dimension in six
dimensions, indicating that replica symmetry breaking occurs above six
dimensions. In two space dimensions, the strong-disorder renormalization group
results for the fractal dimension are in good agreement with essentially exact
numerical results, but the small difference is significant. We discuss the
origin of this close agreement. For the greedy algorithm there is analytical
expectation that the fractal dimension is equal to the space dimension in six
dimensions and our numerical results are consistent with this expectation.
