Timeline for residually finite-by-$\mathbb{Z}$ groups are residually finite
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 23, 2022 at 10:45 | comment | added | ghc1997 | Thanks @Ycor, on a related note, when $G$ is residually finite, but $N$ is not f.g, may I ask if you know any necessary condition that $N$ needs to have (other than being residually finite)? | |
| Aug 23, 2022 at 10:31 | comment | added | YCor | @ghc1997 To weaken the condition that $N$ is fg: if $N$ is RF and has finitely many subgroup of each given finite index (automatic if $N$ is fg), then it is true that $G$ is RF. An even weaker condition is that the $\mathbf{Z}$-action on the set of normal subgroups of finite index of $N$ has no infinite orbit. (Still, this is not yet necessary, as the lampighter group $C_2\wr \mathbf{Z}$ witnesses.) | |
| Aug 23, 2022 at 10:15 | comment | added | ghc1997 | Hi Yves, in the case where $G$ is residually finite, but $N$ is not f.g. Are there any results that tell us what other properties $N$ need to have? As you pointed out, $N$ being residually finite is a necessary but not sufficient condition. | |
| Oct 17, 2011 at 17:37 | history | answered | YCor | CC BY-SA 3.0 |