Some remarks on the homology groups of wreath products

On FC-central extensions of groups of intermediate growth

“…Our goal in this section is to exhibit examples of group of intermediate growth whose center contains Z ∞ . Consider a permutation wreath product W = L ≀ X G. In Tappe [Tap81], a full description of the Schur multiplier of W is given. For a pair (x 1 , x 2 ) ∈ X × X, we say its orbit M = (x 1 , x 2 ) • G under the diagonal action of G on X × X is an orbit of trivial sign, if (x 2 , x 1 ) / ∈ M .…”

“…Denote by t the number of orbits of trivial sign in X × X. Then by [Tap81], the Schur multiplier H 2 (W, Z) contains the direct sum of t copies of H 1 (L) ⊗ H 1 (L).…”

“…By [Tap81], Lemma 7.1 implies that the Schur multiplier H 2 (Z ≀ Xn G n+3 , Z) has a direct summand H 1 (Z) ⊗ H 1 (Z) indexed by the orbit of ((1 n , id), (1 n , ab)). Let [β] be a generator of this summand H 1 (Z) ⊗ H 1 (Z) (as an abelian group), then we can take the central extension of Z ≀ Xn G n+2 corresponding to the 2-cocycle [β].…”