Solvable groups that are simple connected at \(\infty\) (Q1081891)

A connected locally finite CW-complex X, is simply connected at \(\infty\) if for each compact set \(C\subset X\) there is a compact set D(C)\(\subset X\) such that loops in X-D are homotopically trivial in X-C. A finitely presented group, G, is simply connected at \(\infty\) if for some (equivalently any) finite complex Y, with \(\pi _ 1(Y)=G\), the universal cover of Y is simply connected at \(\infty\). If G is simply connected at \(\infty\), then \(H^ 2(G; {\mathbb{Z}}G)=0\). We prove: Theorem 1: Let \(1\to H\to G\to K\to 1\) be a short exact sequence of infinite groups. Assume that G is finitely presented, K is 1-ended, H is abelian and \(h\in H\) has infinite order. Then G is simply connected at \(\infty.\) From this we derive: The main theorem: Let G be a finitely presented solvable group with commutator series \(G\triangleright G^{(1)}\triangleright...\triangleright G^{(n)}\triangleright G^{(n+1)}=1\). If \(G^{(n)}\) contains an element of infinite order, then either G is simply connected at \(\infty\) (and hence \(H^ 2(G; {\mathbb{Z}}G)=0)\) or G contains a normal subgroup, \(\Lambda\), of finite index; and \(\Lambda\) contains a finite normal subgroup, F, such that \(\Lambda\) /F is isomorphic to one of the groups \(<x,y:\) \(x^{-1}yx=y^ p>\). In this case \(H^ 2(G; {\mathbb{Z}}G)\neq 0\) unless \(p=0\).