  The formation of astrophysical and primordial black holes influences the
distribution of dark matter surrounding them. Black holes are thus expected to
carry a dark matter `dress' whose properties depend on their formation
mechanism and on the properties of the environment. Here we carry out a
numerical and analytical study of the merger of dressed black holes, and show
that the distribution of dark matter around them dramatically affects the
dynamical evolution of the binaries. Although the final impact on the merger
rate of primordial black holes is rather small with respect to the case of
`naked' black holes, we argue that our analysis places the calculation of this
rate on more solid ground, with LIGO-Virgo observations ruling out a dark
matter fraction of $10^{-3}$ for primordial black holes of 100 solar masses,
and it paves the way to more detailed analyses of environmental effects induced
by dark matter on the gravitational wave emission of binary black holes.
