On a conjecture of E. Rapaport Strasser about knot-like groups and its pro-p version

A group G is knot-like if it is finitely presented of deficiency 1 and has abelianization image. We prove the conjecture of E. Rapaport Strasser that if a knot-like group G has a finitely generated commutator subgroup G′ then G′ should be free in the special case when the commutator G′ is residually finite. It is a corollary of a much more general result : if G is a discrete group of geometric dimension n with a finite K(G,1)-complex Y of dimension n, Y has Euler characteristics 0, N is a normal residually finite subgroup of G, N is of homological type FPn-1 and image then N is of homological type FPn and hence G/N has finite virtual cohomological dimension vcd(G/N)=cd(G)-cd(N). In particular either N has finite index in G or cd(N)less-than-or-equals, slantcd(G)-1. Furthermore we show a pro-p version of the above result with the weaker assumption that G/N is a pro-p group of finite rank. Consequently a pro-p version of Rapaportʹs conjecture holds.