  Galaxy cluster abundance measurements provide a classic test of cosmology.
They are most sensitive to the evolved amplitude of fluctuations, usually
expressed as $S_8 = \sigma_8\sqrt{\Omega_m/0.3}$. Thus, abundance constraints
exhibit a strong degeneracy between $\sigma_8$ and $\Omega_{\rm m}$, as do
other similar low-redshift tests such as cosmic shear. The mass distribution in
the infall region around galaxy clusters, where material is being accreted from
the surrounding field, also exhibits a cosmological dependence, but in this
case it is nearly orthogonal to the $S_8$ direction in the
$\Omega_m$--$\sigma_8$ plane, making it highly complementary to halo abundance
or cosmic shear studies. We explore how weak lensing measurements of the infall
region might be used to complement abundance studies, considering three
different tests. The splashback radius is a prominent feature of the infall
region; we show that detection of this feature in lensing data from the Euclid
survey could independently constrain $\Omega_{\rm m}$ and $\sigma_8$ to $\pm
0.05$. Another feature, the depletion radius where the bias reaches a minimum,
also shows cosmological dependence, though it is challenging to observe in
practice. The strongest constraints come from direct measurements of the shear
profile in the infall region at $2$--$4\,r_{200{\rm c}}$. Combining the latter
with abundance constraints such as those reported from SRG$/$eROSITA should
reduce the area of the error contours by an estimated factor of $1.2$ using a
sample of clusters observed by the UNIONS survey, or a factor of $3$ using
clusters observed by the Euclid Wide survey over a broader range of redshift.
