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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| Main Author: | |
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| Format: | Article |
| 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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| Summary: | 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. |
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| ISSN: | 1778-3569 |