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The incidence of multiplicity in cool, luminous massive stars is relativelyunknown compared to their hotter counterparts. Here we present radial velocity(RV) measurements and investigate the multiplicity properties of redsupergiants (RSGs) in the 30~Doradus region of the Large Magellanic Cloud. Weprovide absolute RV measurements for our sample and estimate line-of-sightvelocities for the Hodge 301 and SL 639 clusters, which agree well with thoseof hot stars in the same clusters. By combining results for the RSGs with thosefor nearby B-type stars, we estimate systemic velocities and velocitydispersions for the two clusters, obtaining estimates for their dynamicalmasses of $\log (M_{\rm dyn}/M_{\odot})=$3.8$\pm$0.3 for Hodge 301, and anupper limit of $\log (M_{\rm dyn}/M_{\odot})$<3.1$\pm$0.8 for SL 639, assumingVirial equilibrium. Analysis of the multi-epoch data reveals one RV-variable,potential binary candidate (VFTS744), which is likely a semi-regular variableasymptotic giant branch star. We estimate an upper limit on the observed binaryfraction for our sample of 0.3, where we are sensitive to maximum periods forindividual objects in the range of 1 to 10 000 days and mass-ratios above 0.3depending on the data quality. From simulations of the RV measurements frombinary systems given the current data we conclude that systems within theparameter range q>0.3, $\log$P[days]<3.5, would be detected by our variabilitycriteria, at the 90% confidence level. The intrinsic binary fraction,accounting for observational biases, is estimated using simulations of binarysystems with an empirically defined distribution of parameters where orbitalperiods are uniformly distributed in the 3.3<$\log$P[days]<4.3 range. A rangeof intrinsic binary fractions are considered; a binary fraction of 0.3 is foundto best reproduce the observed data. [Abridged]

The VLT-FLAMES Tarantula Survey: XXXI. Radial velocities and multiplicity constraints of red supergiant stars in 30 Doradus