  Multi-object adaptive optics (MOAO) has been demonstrated by the CANARY
instrument on the William Herschel Telescope. However, for proposed MOAO
systems on the next generation Extremely Large Telescopes, such as EAGLE, many
challenges remain. Here we investigate requirements that MOAO operation places
on deformable mirrors (DMs) using a full end-to-end Monte-Carlo AO simulation
code. By taking into consideration a prior global ground-layer (GL) correction,
we show that actuator density for the MOAO DMs can be reduced with little
performance loss. We note that this reduction is only possible with the
addition of a GL DM, whose order is greater than or equal to that of the
original MOAO mirrors. The addition of a GL DM of lesser order does not affect
system performance (if tip/tilt star sharpening is ignored). We also quantify
the maximum mechanical DM stroke requirements (3.5 $\mu$m desired) and provide
tolerances for the DM alignment accuracy, both lateral (to within an eighth of
a sub-aperture) and rotational (to within 0.2$^\circ$). By presenting results
over a range of laser guide star asterism diameters, we ensure that these
results are equally applicable for laser tomographic AO systems. We provide the
opportunity for significant cost savings to be made in the implementation of
MOAO systems, resulting from the lower requirement for DM actuator density.
