Timeline for Quasi-isometry of solvable minimax groups
Current License: CC BY-SA 4.0
9 events
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| Apr 26, 2021 at 7:57 | history | edited | ARG | CC BY-SA 4.0 |
corrected typos and corrected some important points (and tried not to make the comments obsolete in doing so)
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| Apr 21, 2021 at 15:38 | comment | added | ARG | @Ycor Indeed, forgot to mention split sequence, thanks for pointing this out. | |
| Apr 21, 2021 at 15:34 | comment | added | ARG | @Ycor ... to "question in general": well I definitively do not want an answer which is a necessary and sufficient condition (because that would be an obvious open question). [But thanks for stating the actual conjectural status]. I would just like some "simple" known "examples" of minimax solvable groups which are QI. | |
| Apr 21, 2021 at 15:32 | comment | added | YCor | It's not a "construction". But every f.g. virtually solvable minimax group is indeed virtually nilpotent-by-$\mathbf{Z}^d$ for some $d$ (for $d\ge 2$ it's not always virtually such a split extension). | |
| Apr 21, 2021 at 15:30 | comment | added | ARG | @Ycor thanks for all the comments. For (b) I changed the question in the middle of writing and Z was originally Abelian (and not cyclic); this is why (b) is blatantly false now. To (d): thanks for the info! so if I replace Z by an Abelian group, this construction covers all solvable minimax groups? seems strange... (must be missing an important point). To "can the ratios..." I'm not too sure what you mean to say, seems rather to support the fact that the ratios of said powers must be multiple of each other...? | |
| Apr 21, 2021 at 12:58 | comment | added | YCor | Addressing the question in general, a necessary and sufficient condition might be "be cocompact lattices in the same group, that is a virtually solvable Lie group over a finite product of locally compact fields of characteristic zero". This is conjecturally true in the polycyclic case. | |
| Apr 21, 2021 at 12:49 | comment | added | YCor | Addressing the question "can the ratios of the powers...": In the case of SOL, there are non-commensurable lattices, and I think that it precisely comes from the fact that there are matrices in $\mathrm{SL}_2(\mathbf{Z})$ with trace $\ge 3$, which have no conjugate powers; | |
| Apr 21, 2021 at 12:47 | comment | added | YCor | Comments on the remarks: (b) this construction covers nilpotent-by-cyclic solvable minimal groups, arbitrary solvable minimax groups are usually not virtually nilpotent-by-cyclic (even polycyclic ones). (d) actually all f.g. solvable minimax groups are virtually nilpotent-by-abelian. Essentially the restriction in this class of groups is that the nilpotent radical has codimension $\le 1$. This is a strong restriction (for example the polynomial Dehn function / exponential Dehn function / infinitely presented trichotomy is much simpler in this special case). | |
| Apr 21, 2021 at 11:03 | history | asked | ARG | CC BY-SA 4.0 |