  The work discusses equivariant asymptotic dimension (also known as "wide
equivariant covers", "$N$-$\mathcal F$-amenability" or "amenability dimension",
and "$d$-BLR condition") and its generalisation, transfer reducibility, which
are versions of asymptotic dimension invented for the proofs of the
Farrell--Jones and Borel conjectures.
  We prove that groups of null equivariant asymptotic dimension are exactly
virtually cyclic groups. We show that a covering of the boundary always extends
to a covering of the whole compactification. We provide a number of
characterisations of equivariant asymptotic dimension in the general setting of
homotopy actions, including equivariant counterparts of classical
characterisations of asymptotic dimension. Finally, we strengthen the result of
Mole and Rueping (arXiv:1308.2799) about equivariant refinements from finite
groups to infinite groups.
