  We show that the electron mobility in ideal, free-standing two-dimensional
`buckled' crystals with broken horizontal mirror ($\sigma_{\rm h}$) symmetry
and Dirac-like dispersion (such as silicene and germanene) is dramatically
affected by scattering with the acoustic flexural modes (ZA phonons). This is
caused both by the broken $\sigma_{\rm h}$ symmetry and by the diverging number
of long-wavelength ZA phonons, consistent with the Mermin-Wagner theorem.
Non-$\sigma_{\rm h}$-symmetric, `gapped' 2D crystals (such as semiconducting
transition-metal dichalcogenides with a tetragonal crystal structure) are
affected less severely by the broken $\sigma_{\rm h}$ symmetry, but equally
seriously by the large population of the acoustic flexural modes. We speculate
that reasonable long-wavelength cutoffs needed to stabilize the structure
(finite sample size, grain size, wrinkles, defects) or the anharmonic coupling
between flexural and in-plane acoustic modes (shown to be effective in
mirror-symmetric crystals, like free-standing graphene) may not be sufficient
to raise the electron mobility to satisfactory values. Additional effects (such
as clamping and phonon-stiffening by the substrate and/or gate insulator) may
be required.
