  We present a high precision Monte Carlo study of the finite temperature $Z_2$
gauge theory in 2+1 dimensions. The duality with the 3D Ising spin model allows
us to use powerful cluster algorithms for the simulations. For temporal
extensions up to $N_t=16$ we obtain the inverse critical temperature with a
statistical accuracy comparable with the most accurate results for the bulk
phase transition of the 3D Ising model. We discuss the predictions of T. W.
Capehart and M.E. Fisher for the dimensional crossover from 2 to 3 dimensions.
Our precise data for the critical exponents and critical amplitudes confirm the
Svetitsky-Yaffe conjecture. We find deviations from Olesen's prediction for the
critical temperature of about 20%.
