On the Dehn functions of a class of monadic one-relation monoids
We give an infinite family of monoids $\Pi _N$ (for $N=2, 3,\,\dots $), each with a single defining relation of the form $bUa = a$, such that the Dehn function of $\Pi _N$ is at least exponential. More precisely, we prove that the Dehn function $\partial _N(n)$ of $\Pi _N$ satisfies $\partial _N(n)...
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| Language: | English |
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Académie des sciences
2024-09-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.554/ |
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| author | Nyberg-Brodda, Carl-Fredrik |
| author_facet | Nyberg-Brodda, Carl-Fredrik |
| author_sort | Nyberg-Brodda, Carl-Fredrik |
| collection | DOAJ |
| description | We give an infinite family of monoids $\Pi _N$ (for $N=2, 3,\,\dots $), each with a single defining relation of the form $bUa = a$, such that the Dehn function of $\Pi _N$ is at least exponential. More precisely, we prove that the Dehn function $\partial _N(n)$ of $\Pi _N$ satisfies $\partial _N(n) \succeq N^{n/4}$. This answers negatively a question posed by Cain & Maltcev in 2013 on whether every monoid defined by a single relation of the form $bUa=a$ has quadratic Dehn function. Finally, by using the decidability of the rational subset membership problem in the metabelian Baumslag–Solitar groups $\operatorname{BS}(1,n)$ for all $n \ge 2$, proved recently by Cadilhac, Chistikov & Zetzsche, we show that each $\Pi _N$ has decidable word problem. |
| format | Article |
| id | doaj-art-20c6872787be434c80819fd0dc3cf33c |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-20c6872787be434c80819fd0dc3cf33c2025-02-07T11:22:28ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-09-01362G771373010.5802/crmath.55410.5802/crmath.554On the Dehn functions of a class of monadic one-relation monoidsNyberg-Brodda, Carl-Fredrik0Laboratoire d’Informatique Gaspard-Monge, Université Gustave Eiffel (Paris)We give an infinite family of monoids $\Pi _N$ (for $N=2, 3,\,\dots $), each with a single defining relation of the form $bUa = a$, such that the Dehn function of $\Pi _N$ is at least exponential. More precisely, we prove that the Dehn function $\partial _N(n)$ of $\Pi _N$ satisfies $\partial _N(n) \succeq N^{n/4}$. This answers negatively a question posed by Cain & Maltcev in 2013 on whether every monoid defined by a single relation of the form $bUa=a$ has quadratic Dehn function. Finally, by using the decidability of the rational subset membership problem in the metabelian Baumslag–Solitar groups $\operatorname{BS}(1,n)$ for all $n \ge 2$, proved recently by Cadilhac, Chistikov & Zetzsche, we show that each $\Pi _N$ has decidable word problem.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.554/ |
| spellingShingle | Nyberg-Brodda, Carl-Fredrik On the Dehn functions of a class of monadic one-relation monoids Comptes Rendus. Mathématique |
| title | On the Dehn functions of a class of monadic one-relation monoids |
| title_full | On the Dehn functions of a class of monadic one-relation monoids |
| title_fullStr | On the Dehn functions of a class of monadic one-relation monoids |
| title_full_unstemmed | On the Dehn functions of a class of monadic one-relation monoids |
| title_short | On the Dehn functions of a class of monadic one-relation monoids |
| title_sort | on the dehn functions of a class of monadic one relation monoids |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.554/ |
| work_keys_str_mv | AT nybergbroddacarlfredrik onthedehnfunctionsofaclassofmonadiconerelationmonoids |