By Sheppard L.

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43 One-relator presentations [Lyn50]. Let X | r be a one-relator presentation of a group G. 17) where r¯ is a root of r in the free group F with X as basis. Further, if r is not a proper power in the free group F , then π2 (K) = 0. Remark. Let a group G admit an aspherical free presentation. 15) that for any other free presentation 1 → R → F → G → 1, the relation module R/R is a projective Z[G]-module. 44 (M. J. Dunwoody) [Dun72]. Consider the trefoil group G given by the presentation a, b | a2 = b3 .

First let us show that the group G is residually nilpotent. Let R1 be the subgroup in G generated by the central elements βi , i ≥ 1. Then G/R1 is a direct product of nilpotent groups and hence G/R1 is residually nilpotent. Therefore, γω (G) ⊆ R1 . Let 1 = x ∈ γω (G). Then x can be written as x = β1k1 . . 39) for some n ≥ 1 and some integers ki , i = 1, . . , n at at least one ki not equal to zero. Consider the group n −1 Gi )/{ai a−1 i+1 βi βi+1 , i < n}. G(n) = ( ai , i = 1, . . , n ⊕ i=1 There exists the natural epimorphism G → G(n), whose kernel is the subgroup in G generated by elements xi , yi , i > n.

Cm ), which clearly is again an identity sequence. Deﬁne the following operations, called Peiﬀer operations, on the class of identity sequences: (i) replace each wi by any word equal to it in F ; (ii) delete two consecutive terms in the sequence if one is equal identically to the inverse of the other; (iii) insert two consecutive terms in the sequence one of which is equal identically to the inverse of the other; (iv) replace two consecutive terms ci , ci+1 by terms ci+1 , c−1 i+1 ci ci+1 ; (v) replace two consecutive terms ci , ci+1 by terms ci ci+1 c−1 i , ci .

### [Article] Insights on bias and information in group-level studies by Sheppard L.

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