Abstract
In this paper central extensions of free groups of infinitely based varieties of S.I. Adian are constructed. Using this extensions we prove that the Schur multipliers of mentioned relatively free groups are free Abelian groups of infinite rank. It is well-known that these varieties are given by identities in two variables. For a fixed rank $m$, the set of free groups of rank $m$ of these varieties has the cardinality of continuum.
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