In searching for a counterexample in homological stability, I came across the following question:

Is there a known example of a finitely presented group $G$, so that the group ring $\mathbb{Z}[G]$ has infinite Bass stable rank?


Yes, the integral group ring $\mathbb{Z}[F_2]$ of the free group $F_2$ on two generators has infinite stable rank.

This can be deduced from [1, Corollary 3.6]:

There exists a cyclic $\mathbb{Z}[F_2]$-module $M$ with the following property. For every $N \ge 1$, there exists an epimorphism $\theta_N: (\mathbb{Z}[F_2])^N \twoheadrightarrow M$ such that $(\mathbb{Z}[F_2])^N$ cannot be generated by $N$ elements one of which is contained in $\ker \theta_N$.

It should be easy to see that the cyclic $\mathbb{Z}[F_2]$-module $M$ above has infinite stable rank. (Actually, explicit non-stable unimodular rows of length $N$ for every $N \ge 2$ can be extracted from the proof).

Then combine the previous result with [2, Lemma 11.4.6]

If $n$ is in the stable range of a finitely generated right $R$-module $M$ then $n$ is in the stable range of any factor module $M/K$.

to conclude that the stable rank of $\mathbb{Z}[F_2]$ is infinite.

[1] M. Evans, "Presentations of groups involving more generators than necessary", 1992.
[2] J. McConnell and J. Robson, "Noncommutative Noethering rings", 1987.


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