## Abstract

Abstract: An infinite finitely presented nilsemigroup with identity x^{9} = 0 is constructed. This construction answers the question of L.N. Shevrin and M.V. Sapir. The proof is based on the construction of a sequence of geometric complexes, each obtained by gluing several simple 4-cycles (squares). These complexes have certain geometric and combinatorial properties. Actually, the semigroup is the set of word codings of paths on such complexes. Each word codes a path on some complex. Defining relations correspond to pairs of equivalent short paths. The shortest paths in terms of the natural metric are associated with nonzero words in the subgroup. Codings that are not presented by some path or presented by non-shortest paths can be reduced to a zero word.

Original language | English |
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Pages (from-to) | 81-85 |

Number of pages | 5 |

Journal | Doklady Mathematics |

Volume | 101 |

Issue number | 2 |

DOIs | |

State | Published - 1 Mar 2020 |

### Bibliographical note

Publisher Copyright:© 2020, Pleiades Publishing, Ltd.

### Funding

This work was supported by the Russian Science Foundation, grant no. 17-11-01377. The second author is the winner of the contest “Young Russian Mathematics.” ACKNOWLEDGMENTS

Funders | Funder number |
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Russian Science Foundation | 17-11-01377 |

## Keywords

- Burnside-type problems
- finitely presented semigroups