Schur multiplicators of infinite pro-\(p\)-groups with finite coclass. (Q948878)

The author studies the Schur multiplicators \(M(G)\) for infinite pro-\(p\)-groups \(G\) of finite coclass, motivated by the fact that they have a significant influence on the Schur multiplicators of finite \(p\)-groups with the same coclass. For \(p>2\) it is proved that \(M(G)\) is infinite. More precisely the torsion-free rank of \(M(G)\) is equal to \(p^{t-1}(p-1)/2\), where \(t\) is the central exponent of \(G\) defined as follows: if \(N\) is the hypercenter of \(G\) and \(T/N\) is the Fitting subgroup of \(G/N\), then the center \(C/T\) of \(G/T\) is cyclic of order \(p^t\). The theorem is not valid for \(p=2\): infinite pro-2-groups of small coclass are investigated, showing that \(M(G)\) can be infinite, finite or even trivial.