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 G/G′≃ZG/G′≃Z. We prove the conjecture of E. Rapaport Strasser that if a knot-like group G   has a finitely generated commutator subgroup G′G′ then G′G′ should be free in the special case when the commutator G′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)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-1FPn-1 and G/N≃ZG/N≃Z then N   is of homological type FPnFPn and hence G/NG/N has finite virtual cohomological dimension vcd(G/N)=cd(G)-cd(N)vcd(G/N)=cd(G)-cd(N). In particular either N has finite index in G   or cd(N)⩽cd(G)-1cd(N)⩽cd(G)-1.Furthermore we show a pro-p   version of the above result with the weaker assumption that G/NG/N is a pro-p group of finite rank. Consequently a pro-p version of Rapaport's conjecture holds.