A Note on Group Extensions and Proper 3-Realizability

Abstract

The interaction between the study of three-dimensional manifolds and a particular stream of group theory has often been fruitful. In the realm of this, we recall that a finitely presented group G is properly 3-realizable if for some finite 2-dimensional CW-complex X with \({\pi_1(X) \cong G}\), the universal cover of X has the proper homotopy type of a 3-manifold. In this paper, we generalize a previous result on the direct products of groups; more precisely, we show that if \({N \rightarrow G \rightarrow Q}\) is a short exact sequence of infinite finitely presented groups, then G is properly 3-realizable. In particular, any semidirect product of two infinite finitely presented groups is properly 3-realizable. As an application, we show proper 3-realizability for certain classes of groups.