Has the (left, right, 2-sided) noetherian property of the integral group ring of arithmetic groups like $GL\_n(Z)$ been considered in the literature?


Motivation: a recent trend has been to study "representation stability" properties of sequences of groups. The basic property that one establishes along these lines is a sort of noetherian property and a prerequisite for that property is that the group rings are noetherian (so far to my knowledge only sequences of finite groups have been considered).



Does GL_n(Z) have a noetherian group ring?
Obviously a group algebra is left-noetherian iff it's right noetherian, let's call it noetherian (as usual).


If $R[G]$ is left noetherian for some nonzero commutative ring $R$ (associative unital), then $G$ is noetherian, i.e. satisfies the max property for subgroups, i.e. every subgroup is finitely generated, or equivalently every ascending sequence of subgroup stabilizes.


Indeed if $H$ is a subgroup, then the kernel of the $R$-module homomorphism $R[G]\to R[G/H]$ is the left ideal $I\_H$ consisting of finitely supported sums $\sum\alpha\_g\delta\_g$ such that $\sum\_{g\in g\_0H}\alpha\_g=0$ for every left coset $g\_0H$. Since $R\neq 0$ the map $H\mapsto I\_H$ is injective and increasing, whence the result.


Examples of noetherian groups are virtually polycyclic groups, and for them $R[G]$ is noetherian for every noetherian $R$. These are the only known examples with $R[G]$ noetherian (this is a well-known open question).


Still, there exists a few other examples of noetherian groups, first constructed by Olshanskii (Tarski monsters and variants), for which the group algebra is not known to be noetherian.


On the other hand plenty of groups are known not to be noetherian and thus do not have a noetherian group algebra:


* infinitely generated groups;
* groups with a non-abelian free subgroup (they contain a free subgroup on countably many generators), e.g. $\mathrm{GL}(n,\mathbf{Z})$ for all $n\ge 2$;
* elementary amenable (e.g. solvable) groups that are not virtually polycyclic;
* (thanks to the previous 2 items and Tits' alternative): all linear groups that are not virtually polycyclic.


The question about 2-sided noetherianity is a bit more delicate: the obvious obstruction is max-n (maximal condition on normal subgroups). This property fails for many groups (e.g. $\mathrm{GL}(2,\mathbf{Z})$) but holds for many groups (e.g. $\mathrm{GL}(n,\mathbf{Z})$ for $n\ge 3$) and I do not know if their integral group algebra is 2-sided noetherian.




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Edit (Oct. 2020): it's been proved by P. Kropholler and Lorensen [2], strongly relying on work of Bartholdi, that **for every nonzero $R$ and nonamenable $G$, $R[G]$ is not noetherian**. Indeed it is enough to check it for $R$ a domain (or even a field), in which case they observed that part of the proof of Bartholdi's main theorem in [1] shows that for some $n$, $R[G]^{n+1}$ embeds into $R[G]^n$ as left $R[G]$-module (this easily entails non-noetherianity).


All (few) known examples of non-virtually-polycyclic noetherian groups $G$ are either known to be non-amenable (e.g., as consequence of Property T), or just not known to be amenable or not (and probably rather expected to be non-amenable too). Hence, finding non-virtually-polycyclic groups with noetherian group ring seems out of reach at the moment.


[1] L. Bartholdi. Amenability of groups is characterized by Myhill's Theorem (with an appendix by D. Kielak). J. EMS 2019. [ArXiv link](https://arxiv.org/abs/1605.09133)


[2] P. Kropholler, K. Lorensen. Group-graded rings satisfying the strong rank condition. J. Algebra 2019. [ArXiv link](https://arxiv.org/abs/1901.10001)


