Skip to main content
Correcting title
Source Link
ADL
  • 2.8k
  • 1
  • 24
  • 32

$\mathbb residually finite-by-$\mathbb{Z}$-by-residually finite groups are residually finite

I believe I read somewhere that $\mathbb{Z}$residually finite-by-residually finite$\mathbb{Z}$ groups are residually finite. That is, if $G/N$$N$ is residually finite wherewith $N\cong \mathbb{Z}$$G/N\cong \mathbb{Z}$ then $G$ is residually finite.

However, I cannot remember where I read this, and nor can I find another place which says it. I was therefore wondering if someone could confirm whether this is true or not, and if it is give either a proof or a reference for this result? (If not, a counter-example would not go amiss!)

Note that I definitely know it is true if $G/N$$N$ is f.g. free (there isthis can be found in a paper of BG. Baumslag and Levin which mentions this, called "A class"Finitely generated cyclic extensions of one-relatorfree groups with torsion"are residually finite" (Bull. Amer. Math. Soc., if I remember correctly5, 87-94, 1971)).

$\mathbb{Z}$-by-residually finite groups are residually finite

I believe I read somewhere that $\mathbb{Z}$-by-residually finite groups are residually finite. That is, if $G/N$ is residually finite where $N\cong \mathbb{Z}$ then $G$ is residually finite.

However, I cannot remember where I read this, and nor can I find another place which says it. I was therefore wondering if someone could confirm whether this is true or not, and if it is give either a proof or a reference for this result? (If not, a counter-example would not go amiss!)

Note that I definitely know it is true if $G/N$ is f.g. free (there is a paper of B. Baumslag and Levin which mentions this, called "A class of one-relator groups with torsion", if I remember correctly).

residually finite-by-$\mathbb{Z}$ groups are residually finite

I believe I read somewhere that residually finite-by-$\mathbb{Z}$ groups are residually finite. That is, if $N$ is residually finite with $G/N\cong \mathbb{Z}$ then $G$ is residually finite.

However, I cannot remember where I read this, and nor can I find another place which says it. I was therefore wondering if someone could confirm whether this is true or not, and if it is give either a proof or a reference for this result? (If not, a counter-example would not go amiss!)

Note that I definitely know it is true if $N$ is f.g. free (this can be found in a paper of G. Baumslag, "Finitely generated cyclic extensions of free groups are residually finite" (Bull. Amer. Math. Soc., 5, 87-94, 1971)).

Source Link
ADL
  • 2.8k
  • 1
  • 24
  • 32

$\mathbb{Z}$-by-residually finite groups are residually finite

I believe I read somewhere that $\mathbb{Z}$-by-residually finite groups are residually finite. That is, if $G/N$ is residually finite where $N\cong \mathbb{Z}$ then $G$ is residually finite.

However, I cannot remember where I read this, and nor can I find another place which says it. I was therefore wondering if someone could confirm whether this is true or not, and if it is give either a proof or a reference for this result? (If not, a counter-example would not go amiss!)

Note that I definitely know it is true if $G/N$ is f.g. free (there is a paper of B. Baumslag and Levin which mentions this, called "A class of one-relator groups with torsion", if I remember correctly).