Second Cohomology and Nilpotency Class 2

Conditions are given for a class 2 nilpotent group to have no central extensions of class 3. This is related to Betti numbers and to the problem of representing a class 2 nilpotent group as the fundamental group of a smooth projective variety. Surveys of the work on the characterization of the fundamental groups of smooth projective varieties and Kähler manifolds (see [1],[3], [9]) indicate that torsion-free nilpotent groups have been both attractive and problematic in this context. For example, it took a long time to show, contrary to the beliefs of many researchers in the area, that the fundamental group of a smooth projective variety can be non-abelian torsion-free nilpotent ([2], [11]). Even more interestingly from an algebraist’s point of view, nilpotent groups of class 2 have played a significant role as test cases. There are structural reasons for this. Let P denote the class of groups isomorphic to fundamental groups of smooth projective varieties. Let γk(G) denote the k-th term of the lower central series of the group G. For a subgroup K of G, let √ K denote the subgroup generated by {g ∈ G | (∃n ∈ Z) g ∈ K}. Due to the work of Deligne [4], Hain [6] and others, it is known that if G ∈ P then the quotients G/ √ γ2(G) and G/ √ γ3(G) in a certain sense determine G/ √ γk(G) for all k. Consequently, if G ∈ P is torsion-free nilpotent of class 2, its class 3 extensions are limited by the underlying geometry. For this and other reasons, class 2 nilpotent groups have generated a certain amount of interest among geometers. For instance, Hain [7] defines groups of Heisenberg type as finitely generated nontrivial central extensions of Z by a torsionfree Abelian group, and demonstrates that fundamental groups of links of isolated singularities of n-dimensional complex algebraic varieties are of Heisenberg type (hence torsion-free and nilpotent of class 2). The definition generalizes the standard Heisenberg groups H2m+1, given by generators {x0, x1, . . . , x2m} and relations (xi, xm+i) = x0 for i = 1, . . . ,m, and (xj , xk) = 1 for all other commutators. These 2000 Mathematics Subject Classification: Primary 20J06, 57T10; Secondary 20F18, 57M05.