= Golod-Shafarevich infinitude criterion for pro-p groups
{c}
If a pro-$p$ presentation on $d$ generators and possibly infinitely many relators $r_i$ satisfies
$$
1-d\tau+\sum_i\tau^{D(r_i)}<0\qquad(0<\tau<1),
$$
with the sum convergent and $D$ the <Magnus degree of a free pro-p group element>, then the presented <pro-p group> is infinite. The Golod-Shafarevich inequality forces its completed group algebra to be infinite-dimensional, which would be impossible for a finite group. This formulation applies to relators whose degrees tend to infinity and is suitable for enumerated power-relation constructions.
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