# Non finitely based varieties of groups defined by finitely many variables

A set $\Sigma$ of group identities is called bounded if there is $n\geq 1$ such that for any $(w\approx 1)\in \Sigma$, we have $w\in F(x_1, \ldots, x_n)$. A variety $\mathbf{V}$ is called bounded defined if $\mathbf{V}=Mod(\Sigma)$ for some bounded set $\Sigma$.

Question: Is there an example of a bounded defined non-finitely based variety of groups?

P.S. $F(x_1, \ldots, x_n)$ is the free group of rank $n$.

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I think there is a better way to ask this question. Let T be a "narrow" equational theory, with closure of T logically equivalent to closure of S, where S is an equational basis in group theory at most n distinct variables. Does cl(T) have a finite basis necessarily? –  The Masked Avenger Jan 19 at 4:29

Actually, one of the first examples of non-finitely based variety of groups has axiomatic rank two; Adian proved that the following set of identities is independent: $$\{[x^{pn},y^{pn}]^n=1\;|\; p \hbox{ is prime}\},$$ where $n\ge 1003$ is any given odd number. Independence means that none of the laws is a consequence of the other ones.