Corpus ID: 210932690

Computing commutator length is hard

  title={Computing commutator length is hard},
  author={Nicolaus Heuer},
  journal={arXiv: Group Theory},
The commutator length $cl_G(g)$ of an element $g \in [G,G]$ in the commutator subgroup of a group $G$ is the least number of commutators needed to express $g$ as their product. If $G$ is a non-abelian free groups, then given an integer $n \in \mathbb{N}$ and an element $g \in [G,G]$ the decision problem which determines if $cl_G(g) \leq n$ is NP-complete. Thus, unless P=NP, there is no algorithm that computes $cl_G(g)$ in polynomial time in terms of $|g|$, the wordlength of $g$. This statement… Expand
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