2-knots and their groups by Jonathan A. Hillman
By Jonathan A. Hillman
To assault definite difficulties in four-dimensional knot idea the writer attracts on quite a few concepts, concentrating on knots in S^T4, whose basic teams comprise abelian common subgroups. Their type comprises the main geometrically beautiful and top understood examples. furthermore, it really is attainable to use fresh paintings in algebraic ways to those difficulties. New paintings in 4-dimensional topology is utilized in later chapters to the matter of classifying 2-knots.
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Additional info for 2-knots and their groups
Theorem 2 Let a be a finitely generated group of cohomoJogicaJ dimen- sion 2 with a nontrivial abelian normal subgroup A. Then either A is infinite cyclic or a is solvable. If 0/0' ;:;- Z then either a ;:;- ¢> or A 40 Localization and Asphericily is central and G' is Proof Suppose free; in either case G has deficiency 1. that A is not then Aut(A) is abelian, so G' = cyclic. 8]. G - A infinite ~ CG(A) and is metabelian. Otherwise G CG(A) ~ Z2. As CG(A) with an element of G infinite order modulo CG(A) would generate a subgroup of cohomological dimension 3, which is impossible, GICG(A) must be a torsion group.
Therefore the commutator subgroup of the factors is a free product of finite groups and PDt- groups, and is never a nontrivial free group. (Thus if ITT: rK and not Zit has cohomological dimension 4). Since K ITT: r is torsion free has a cen tral ele- ment of infinite order, Corollary 1 implies that it cannot have deficiency 1, and so in particular T: rK cannot be a nontrivial homotopy ribbon 2-knot (cf. [Co 1983]). Some of the arguments of the next few chapters may be seen in microcosm in next theorem.
This strategy works in considerably greater generality, provided we forgo groups some have information abelian normal about torsion. subgroups, For instance, although there may be finitely solvable presentable infinite solvable groups in which no such subgroup is torsion free. In order to get around this problem we may factor out the maximal locally-finite normal subgroup. (This idea is due to Kropholler). The quotient of a 2-knot group by such a subgroup is then usually a PDt-group over Q. Rosset's Lemma The keystone of the argument of this chapter (and hence of the whole book) is the following lemma of Rosset.