  The two-dimensional Holstein-Hubbard model is studied by means of
continuous-time quantum Monte Carlo simulations. Using
renormalization-group-invariant correlation ratios and finite-size
extrapolation, the critical temperature of the charge-density-wave transition
is determined as a function of coupling strength, phonon frequency, and Hubbard
repulsion. The phase transition is demonstrated to be in the universality class
of the two-dimensional Ising model and detectable via the fidelity
susceptibility. The structure of the ground-state phase diagram and the
possibility of a bipolaronic metal with a single-particle gap above $T_c$ are
explored.
