The finiteness-length formula for Abels groups over infinitely generated rings

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Let RR be a finitely generated commutative ring with unity that is infinitely generated as a \b7Z\b7\mathbb{Z}-module. For n\b7≥2n\b7\geq 2, let \b7An(R)\b7\mathbf{A}_n(R) be the corresponding Abels group and let \b7bphi\b7bphi denote finiteness length; write \b7B2∘(R)\b7\mathbf{B}_2^{\circ}(R) for the indicated subgroup. Finiteness-length formula.

ϕ(An(R))=min⁡{n−2,ϕ(B2∘(R))}for all n≥2.\phi(\mathbf{A}_n(R))=\min\left\{n-2,\phi(\mathbf{B}_2^{\circ}(R))\right\}\quad\text{for all }n\geq 2.

This would determine the finiteness length of these Abels groups in terms of the rank parameter nn and that of the corresponding subgroup. The surrounding discussion presents it as a natural problem and cites lower-bound results and examples, but does not state that the formula has been proved or disproved.

References

Primary source

Yuri Santos Rego, “On the finiteness length of some soluble linear groups”, arXiv:1901.06704 (2021).

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