Abstract
AbstractWe describe an algorithm for computing successive quotients of the Schur multiplierM(G) of a groupGgiven by an invariant finiteL-presentation. As applications, we investigate the Schur multipliers of various self-similar groups, including the Grigorchuk super-group, the generalized Fabrykowski–Gupta groups, the Basilica group and the Brunner–Sidki–Vieira group.
Highlights
The Schur multiplier M (G) of a group G can be defined as the second homology group H2(G, Z)
The Schur multiplier found applications by virtue of the Hopf formula: if F is a free group and R is a normal subgroup of F such that G ∼= F/R, the Schur multiplier of G is isomorphic to the factor group (R ∩ F )/[R, F ]
The Hopf formula yields that every finitely presentable group has a finitely generated Schur multiplier. This fact is used in [15] to prove that the Grigorchuk group is not finitely presentable, as its Schur multiplier is infinitely generated 2-elementary abelian, which answers the questions raised in [6, 25]
Summary
The Schur multiplier M (G) of a group G can be defined as the second homology group H2(G, Z). The Hopf formula yields that every finitely presentable group has a finitely generated Schur multiplier. Besides the Grigorchuk group, there are various examples of self-similar groups for which it is not known whether their Schur multiplier is finitely generated or whether the groups are finitely presented. The first aim of this paper is to introduce an algorithm for investigating the Schur multiplier of a self-similar group, with a view towards demonstrating its finite generation. – For c 4, Mc(BSV) has the form Z2 × Bc, where Bc is an abelian 2-group of rank log2(c/5) + log2(c/9) + 3 and exponent 22 (c−4)/2 +1. All of these groups have an infinitely generated Schur multiplier and are not finitely presentable.
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