Transfinite Lower Central Series of Groups: Parafree Properties and Topological Applications

R. V. Mikhailov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 251-267

Résumé

2-Generated groups $G(i)$, $i=1,2,\dots$, such that $\gamma _{\omega}G(i)\neq\gamma _{\omega+1}G(i)$ are constructed. Moreover, the natural homomorphism $F_2\to G(i)$ of a free group of rank 2 induces isomorphisms $F_2/\gamma _kF_2\simeq G(i)/\gamma _k G(i)$ for all $k\leq q^{i-1}$, where $q$ is a certain prime, and the group $G(1)$ is finitely presented. Methods for realizing a generalized torsion by means of fundamental groups of complements to links are also considered. Torsion-free fundamental groups of 3-manifolds for which the lower central series do not stabilize at the $\omega$th step are constructed. For an arbitrary finite 2-torsion-free abelian group $A$, a 3-manifold (with boundary) is constructed such that $\gamma _{\omega }/\gamma _{\omega +1}\simeq A$ for its fundamental group.