LSST will open new vistas for cosmology in the next decade, but it cannot reach its full potential without data from other telescopes. Cosmological constraints can be greatly enhanced using wide-field ($>20$ deg$^2$ total survey area), highly-multiplexed optical and near-infrared multi-object spectroscopy (MOS) on 4-15m telescopes. This could come in the form of suitably-designed large surveys and/or community access to add new targets to existing projects. First, photometric redshifts can be calibrated with high precision using cross-correlations of photometric samples against spectroscopic samples at $0 < z < 3$ that span thousands of sq. deg. Cross-correlations of faint LSST objects and lensing maps with these spectroscopic samples can also improve weak lensing cosmology by constraining intrinsic alignment systematics, and will also provide new tests of modified gravity theories. Large samples of LSST strong lens systems and supernovae can be studied most efficiently by piggybacking on spectroscopic surveys covering as much of the LSST extragalactic footprint as possible (up to $sim20,000$ square degrees). Finally, redshifts can be measured efficiently for a high fraction of the supernovae in the LSST Deep Drilling Fields (DDFs) by targeting their hosts with wide-field spectrographs. Targeting distant galaxies, supernovae, and strong lens systems over wide areas in extended surveys with (e.g.) DESI or MSE in the northern portion of the LSST footprint or 4MOST in the south could realize many of these gains; DESI, 4MOST, Subaru/PFS, or MSE would all be well-suited for DDF surveys. The most efficient solution would be a new wide-field, highly-multiplexed spectroscopic instrument in the southern hemisphere with $>6$m aperture. In two companion white papers we present gains from deep, small-area MOS and from single-target imaging and spectroscopy.
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