Elementary amenable groups and 4-manifolds with Euler characteristic 0

Abstract

We extend earlier work relating asphericity and Euler characteristics for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups. In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical. Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends or is virtually an extension of Z by a subgroup of Q, or the manifold is aspherical and the group is virtually poly-Z of Hirsch length 4.

Elementary amenable groups of finite hirsch length are locally-finite by virtually-solvable

“…Then by [3,Lemma 1] there are normal subgroups K < H in G such that G/H is finite, H/K is free abelian of rank r > 1 and the action of G/H on H/K by conjugation is effective.…”

“…Since h{K) = h{G) -r<h the inductive hypothesis applies for K, so it has a normal subgroup L containing In general let {G ( \i in /} be the set of finitely generated subgroups of G. Since G t is a finitely generated elementary amenable group with Hirsch length at most h + 1 we see from the previous paragraph that G t has a normal subgroup 77 ( PROOF. If H is such an elementary amenable normal subgroup then it has finite cohomological dimension and so h{H) < oo by [3,Lemma 2]. Moreover H is torsion free, so by the theorem it is virtually solvable.…”

“…However the main result of [5] is needed to prove the converse, stated in [3], that if X is a [G, m]-complex with x(X) -0 and G = n x (X) has a nontrivial torsion free elementary amenable normal subgroup then X is aspherical.…”

“…It clearly contains all locally finite groups and solvable groups, and hence contains all locally finite by virtually solvable groups. In [3] the notion of Hirsch length (as a measure of the size of a solvable group) was extended to this class and it was shown that elementary amenable groups of Hirsch length at most 3 are locally finite by solvable while in general the Hirsch length is bounded above by the rational cohomological dimension. Here we shall show that every elementary amenable group of finite Hirsch length is locally finite by virtually solvable.…”

“…We recall briefly the description of the class of elementary amenable groups given in [4] and the definition of Hirsch length given in [3]. If 3?…”