  We consider the $h$-version of the finite-element method, where accuracy is
increased by decreasing the meshwidth $h$ while keeping the polynomial degree
$p$ constant, applied to the Helmholtz equation. Although the question "how
quickly must $h$ decrease as the wavenumber $k$ increases to maintain
accuracy?" has been studied intensively since the 1990s, none of the existing
rigorous wavenumber-explicit analyses take into account the approximation of
the geometry. In this paper we prove that for nontrapping problems solved using
straight elements the geometric error is order $kh$, which is then less than
the pollution error $k(kh)^{2p}$ when $k$ is large; this fact is then
illustrated in numerical experiments. More generally, we prove that, even for
problems with strong trapping, using degree four (in 2-d) or degree five (in
3-d) polynomials and isoparametric elements ensures that the geometric error is
smaller than the pollution error for most large wavenumbers.
