Values of Rokhlin dimension for actions of compact groups

We show that any finite group admits actions on simple AF algebras with unique trace which have arbitrarily large finite values of Rokhlin dimension with commuting towers. We show similar results for actions of compact Lie groups, with AH algebras with no dimension growth in place of AF algebras. We also relate Rokhlin dimension to the G-index for actions of compact Lie groups on commutative C*-algebras.