KiDS-1000 Cosmology: constraints beyond flat $\Lambda$CDM
  We present constraints on extensions to the flat $\Lambda$CDM cosmological
model by varying the spatial curvature $\Omega_K$, the sum of the neutrino
masses $\sum m_\nu$, the dark energy equation of state parameter $w$, and the
Hu-Sawicki $f(R)$ gravity $f_{R0}$ parameter. With the combined $3\times2$pt
measurements of cosmic shear from the Kilo-Degree Survey (KiDS-1000), galaxy
clustering from the Baryon Oscillation Spectroscopic Survey (BOSS), and
galaxy-galaxy lensing from the overlap between KiDS-1000, BOSS, and the
spectroscopic 2-degree Field Lensing Survey (2dFLenS), we find results that are
fully consistent with a flat $\Lambda$CDM model with
$\Omega_K=0.011^{+0.054}_{-0.057}$, $\sum m_\nu<1.76$ eV (95% CL), and
$w=-0.99^{+0.11}_{-0.13}$. The $f_{R0}$ parameter is unconstrained in our fully
non-linear $f(R)$ cosmic shear analysis. Considering three different model
selection criteria, we find no clear preference for either the fiducial flat
$\Lambda$CDM model or any of the considered extensions. Besides extensions to
the flat $\Lambda$CDM parameter space, we also explore restrictions to common
subsets of the flat $\Lambda$CDM parameter space by fixing the amplitude of the
primordial power spectrum to the Planck best-fit value, as well as adding
external data from supernovae and lensing of the CMB. Neither the
beyond-$\Lambda$CDM models nor the imposed restrictions explored in this
analysis are able to resolve the $\sim 3\sigma$ tension in $S_8$ between the
$3\times2$pt constraints and Planck, with the exception of $w$CDM, where the
$S_8$ tension is resolved. The tension in the $w$CDM case persists, however,
when considering the joint $S_8$-$w$ parameter space. The joint flat
$\Lambda$CDM CMB lensing and $3\times2$pt analysis is found to yield tight
constraints on $\Omega_{\rm m}=0.307^{+0.008}_{-0.013}$,
$\sigma_8=0.769^{+0.022}_{-0.010}$, and $S_8=0.779^{+0.013}_{-0.013}$.
