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We performed a detailed study of the evolution of the star formation rate(SFR) and stellar mass of the brightest group galaxies (BGGs) and theirrelative contribution to the total baryon budget within $R_{200}$($f^{BGG}_{b,200}$). The sample comprises 407 BGGs selected from X-ray galaxygroups ($M_{200}=10^{12.8}-10^{14} \;M_{\odot}$) out to $z\sim1.3$ identifiedin the COSMOS, XMM-LSS, and AEGIS fields. We find that BGGs constitute twodistinct populations of quiescent and star-forming galaxies and their mean SFRis $\sim2$ dex higher than the median SFR at $ z<1.3 $. Both the mean and themedian SFRs decline with time by $>2$ dex. The mean (median) of stellar mass ofBGGs has grown by $0.3$ dex since $z=1.3$ to the present day. We show that upto $\sim45\% $ of the stellar mass growth in a star-forming BGG can be due toits star-formation activity. With respect to $f^{BGG}_{b,200}$, we find it toincrease with decreasing redshift by $\sim0.35$ dex while decreasing with halomass in a redshift dependent manner. We show that the slope of the relationbetween $f^{BGG}_{b,200}$ and halo mass increases negatively with decreasingredshift. This trend is driven by an insufficient star-formation in BGGs,compared to the halo growth rate. We separately show the BGGs with the 20\%highest $f^{BGG}_{b,200}$ are generally non-star-forming galaxies and grow inmass by processes not related to star formation (e.g., dry mergers and tidalstriping). We present the $ M_\star-M_h $ and $ M_\star/M_h-M_h $ relations andcompare them with semi-analytic model predictions and a number of results fromthe literature. We quantify the intrinsic scatter in stellar mass of BGGs atfixed halo mass ($\sigma_{log M_{\star}}$) and find that $\sigma_{logM_{\star}}$ increases from 0.3 dex at $ z\sim0.2 $ to 0.5 dex at $ z\sim1.0 $due to the bimodal distribution of stellar mass.

Brightest group galaxies-II: the relative contribution of BGGs to the total baryon content of groups at z<1.3