  We apply strong-coupling expansion techniques to finite-temperature lattice
pure gauge theory, obtaining dimensionally reduced $Z_N$-symmetric effective
theories. The analytic mappings between the effective couplings and the
original one, viz. $\beta$, allow to estimate the transition point $\beta_c$ of
the 4D theory for a large range of the imaginary-time extent $N_\tau$ of the
lattice. We study the models for SU(3) via Monte Carlo simulation, finding
satisfactory agreement with the critical point of the original theories
especially at low $N_\tau$. We have fixed an error in the group measure used in
arXiv:1010.0951 and provide here the correct numerical results.
