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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).

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I believe the only known examples of groups with Noetherian group rings are polycyclic-by-finite groups. –  arsmath May 3 at 5:04
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see also math.stackexchange.com/questions/344403/… –  YCor May 3 at 9:21

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up vote 19 down vote accepted

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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Minor edits to last sentence. –  Jim Humphreys May 3 at 19:31

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