On the Dehn functions of a class of monadic one-relation monoids

Abstract

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 ${\partial }_N(n)⪰N^{n/4}$. 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 $\mathrm{BS}(1,n)$ for all $n\ge 2$, proved recently by Cadilhac, Chistikov & Zetzsche, we show that each ${\Pi }_N$ has decidable word problem.

arXiv comment19 pages. Comments welcome!