  We report numerical results for the single-hole properties in the $t$-$J$
model and the strong-coupling approximation to the Hubbard model in two
dimensions. Using the hopping basis with over $10^6$ states we discuss (for an
infinite system) the bandwidth, the leading Fourier coefficients in the
dispersion, the band masses, and the spin-spin correlations near the hole. We
compare our results with those obtained by other methods. The band minimum is
found to be at ($\pi/2,\pi/2$) for the $t$-$J$ model for $0.1 \leq t/J \leq
10$, and for the strong-coupling model for $1 \leq t/J \leq 10$. The bandwidth
in both models is approximately $2J$ at large $t/J$, in rough agreement with
loop-expansion results but in disagreement with other results. The
strong-coupling bandwidth for $t/J\agt6$ can be obtained from the $t$-$J$ model
by treating the three-site terms in first-order perturbation theory. The
dispersion along the magnetic zone face is flat, giving a large
parallel/perpendicular band mass ratio.
