Golod–Shafarevich theorem

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In mathematics, the Golod–Shafarevich theorem is a result in non-commutative homological algebra which solves the class field tower problem by showing that class field towers can be infinite. It is named after Evgeny Golod and Igor Shafarevich, who proved it in 1964.

Statement of the inequality

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 ${\displaystyle i\geq 2}$.

Applications

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.

References