An explicit construction of a limit variety of groups

Zidong Gao; Yilin Zhou

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arXiv · 2610.17086

An explicit construction of a limit variety of groups

Zidong Gao·Yilin Zhou

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Abstract

We construct an explicit countable group $G_*=N(R)\rtimes A$ whose generated variety is a limit variety of groups: it has no finite identity basis, whereas every proper subvariety is finitely based. Here $A$ is relatively free in the variety of groups of nilpotency class at most two and exponent dividing four, and $R$ is an explicitly defined quotient of the group algebra $\mathbb F_2[A]$. The proof establishes an ascending chain condition for coefficient spaces indexed by six ordered commutator entries and reduces every proper subvariety to a hereditarily finitely based approximation. Non-finite basability is obtained by transferring independent identities from the constructions of Gupta and Krasilnikov. The resulting variety is contained in $\mathfrak B_4\mathfrak B_2$. This construction provides the first example of such a variety in which both the generators and a basis are explicitly given.

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BibTeX RIS

Zidong Gao, Yilin Zhou. 2026-10-12. An explicit construction of a limit variety of groups. https://arxiv.org/abs/2610.17086

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