  Cluster strong lensing cosmography is a promising probe of the background
geometry of the Universe and several studies have emerged, thanks to the
increased quality of observations using space and ground-based telescopes. For
the first time, we use a sample of five cluster strong lenses to measure the
values of cosmological parameters and combine them with those from classical
probes. In order to assess the degeneracies and the effectiveness of
strong-lensing cosmography in constraining the background geometry of the
Universe, we adopt four cosmological scenarios. We find good constraining power
on the total matter density of the Universe ($\Omega_{\rm m}$) and the equation
of state of the dark energy parameter $w$. For a flat $w$CDM cosmology, we find
$\Omega_{\rm m} = 0.30_{-0.11}^{+0.09}$ and $w=-1.12_{-0.32}^{+0.17}$ from
strong lensing only. Interestingly, we show that the constraints from the
Cosmic Microwave Background (CMB) are improved by factors of 2.5 and 4.0 on
$\Omega_{\rm m}$ and $w$, respectively, when combined with our posterior
distributions in this cosmological model. In a scenario where the equation of
state of dark energy evolves with redshift, the strong lensing constraints are
compatible with a cosmological constant (i.e. $w=-1$). In a curved cosmology,
our strong lensing analyses can accommodate a large range of values for the
curvature of the Universe of $\Omega_{\rm k}=0.28_{-0.21}^{+0.16}$. In all
cosmological scenarios, we show that our strong lensing constraints are
complementary and in good agreement with measurements from the CMB, baryon
acoustic oscillations and Type Ia supernovae. Our results show that cluster
strong lensing cosmography is a potentially powerful probe to be included in
the cosmological analyses of future surveys.
