= p-deficiency
{title2=$\operatorname{def}_p\langle X\mid R\rangle$}
For a <group presentation> $\mathcal P=\langle X\mid R\rangle$ with finitely many generators and a <prime number> $p$, define
$$
\nu_p(r)=\max\{k\geq0:r=w^{p^k}\text{ in }F(X)\},\qquad\operatorname{def}_p(\mathcal P)=|X|-\sum_{r\in R}p^{-\nu_p(r)}.
$$
This uses the unshifted convention. Identity relators have zero weight, and a divergent sum gives value $-\infty$. Some literature subtracts one instead; inequalities must be shifted accordingly. Taking roots in the <free group> matters: roots appearing only after passing to the quotient do not change the relator weight. The <p-rank of a group> bounds this quantity above, and the <index-p rewriting bound for p-deficiency> gives the infinitude criterion <p-deficiency at least one implies infinitude>.
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