Homological properties of parafree Lie algebras

The homology of the lamplighter Lie algebra

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We show that the associated Lie algebra of the Malcev Q-completion of the lamplighter group is the pronilpotent completion of the lamplighter Lie algebra. We also prove that the homology of this completed Lie algebra is of uncountable dimension on each degree.Consider the integer lamplighter group G which can also be regarded as the restricted wreath product Z ≀ Z of two infinite cyclic groups and can be presented asThe explicit description of both, the pronilpotent and Malcev Q-completion of G, is the cornerstone of the recent work of Sergei Ivanov and Roman Mikhailov on the homology of the completion of free groups ([6]).On the Lie side, we use the term lamplighter Lie algebra to name the Lie analogue of G (over any commutative ring R) introduced in [7] as L R = R[x] ⋊ Rt, the semidirect product of the abelian Lie algebras R[x] and Rt with [p, t] = x • p for any polynomial p. In [7] the authors prove that the degree 2 homology H 2 ( L R ; R) of the pronilpotent completion of L R is uncountable. This is then used as an important step in the proof of the non-countability dimension of H 2 (F R ; R) where F R denotes the pronilpotent completion of the R-free Lie algebra on two generators.In this text we first prove:Theorem 1.

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“…On the Lie side, we use the term lamplighter Lie algebra to name the Lie analogue of G (over any commutative ring R) introduced in [7] as L R = R[x] ⋊ Rt, the semidirect product of the abelian Lie algebras R[x] and Rt with [p, t] = x • p for any polynomial p. In [7] the authors prove that the degree 2 homology H 2 ( L R ; R) of the pronilpotent completion of L R is uncountable. This is then used as an important step in the proof of the non-countability dimension of H 2 (F R ; R) where F R denotes the pronilpotent completion of the R-free Lie algebra on two generators.…”