Is there any relation, either conjectural or known, between the Borel regulator for the Quillen $K$theory of algebraic number rings, and the BlochBeilinson regulator from motivic cohomology to real Deligne cohomology?
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1$\begingroup$ What about the book of Burgos Gil? icmat.es/miembros/burgos/files/brbr.pdf $\endgroup$– Matthias WendtCommented Jan 11, 2018 at 9:30

$\begingroup$ probably this helping you to predict the relationship you want :faculty.math.illinois.edu/~dan/KtheoryHandbookforauthorsonly/… $\endgroup$– zeraoulia rafikCommented Jan 11, 2018 at 11:53
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1 Answer
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To elaborate on my comment, the comparison between the regulators of Beilinson and Borel can be found in the book
 J.I. Burgos Gil. The regulators of Beilinson and Borel. CRM Monograph Series, 15. Amer. Math. Soc., 2002. (link to book)
The main result is that ${\rm reg}_{\rm Borel}=2{\rm reg}_{\rm Beilinson}$. A short discussion of the history of the comparison and earlier work can be found on p. 3 of the book. (As a side note, let me point out that $H^1(X,\mathbb{Q}(n))=K_{2n1}(X)$ for $X$ the spectrum of a number field or number ring, so there is no real need to talk about motivic cohomology in this case.)