Golod–Shafarevich theorem

Let ${\displaystyle A=K\langle x_{1},\dots ,x_{n}\rangle }$ be the free algebra over a field ${\displaystyle K}$ in ${\displaystyle n=d+1}$ non-commuting variables, and let ${\displaystyle J}$ be the two-sided ideal of ${\displaystyle A}$ generated by homogeneous elements ${\displaystyle f_{j}}$ of ${\displaystyle A}$ of degree ${\displaystyle d_{j}}$ with

${\displaystyle 2\leq d_{1}\leq d_{2}\leq \cdots }$

where ${\displaystyle d_{j}}$ tends to infinity. Let ${\displaystyle r_{i}}$ be the number of ${\displaystyle d_{j}}$ equal to ${\displaystyle i}$, for all ${\displaystyle i\geq 2}$.

Let ${\displaystyle B=A/J}$; this is a graded algebra. Let ${\displaystyle b_{j}=\dim B_{j}}$. The fundamental inequality of Golod and Shafarevich states that

${\displaystyle b_{j}\geq nb_{j-1}-\sum _{i=2}^{j}b_{j-i}r_{i}.}$

for all ${\displaystyle j}$. As a consequence, ${\displaystyle B}$ is infinite-dimensional if ${\displaystyle r_{i}\leq d^{2}/4}$ for all ${\displaystyle i\geq 2}$.

This result has important applications in combinatorial group theory:

• If ${\displaystyle G}$ is a non-trivial finite p-group, then ${\displaystyle r>d^{2}/4}$, where ${\displaystyle d=\dim H^{1}(G,\mathbb {Z} /p\mathbb {Z} )}$ and ${\displaystyle r=\dim H^{2}(G,\mathbb {Z} /p\mathbb {Z} )}$ (the mod ${\displaystyle p}$ cohomology groups of ${\displaystyle G}$). In particular, if ${\displaystyle G}$ is a finite ${\displaystyle p}$-group with minimal number of generators ${\displaystyle d}$ and has ${\displaystyle r}$ relators in a given presentation, then ${\displaystyle r>d^{2}/4}$.

• For each prime ${\displaystyle p}$, there is an infinite group ${\displaystyle G}$ generated by three elements in which each element has order a power of ${\displaystyle p}$. The group ${\displaystyle G}$ provides a counterexample to the generalised Burnside conjecture: it is a finitely generated infinite torsion group, although there is no uniform bound on the order of its elements.