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

We give an infinite family of monoids ${\mathrm{\Pi }}_N$ (for $N=2,3,\dots$), each with a single defining relation of the form $bUa=a$, such that the Dehn function of ${\mathrm{\Pi }}_N$ is at least exponential. More precisely, we prove that the Dehn function ${\mathrm{\partial }}_N(n)$ of ${\mathrm{\Pi }}_N$ satisfies ${\mathrm{\partial }}_N(n)⪰N^{n\mathrm{/}4}$. 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 ${\mathrm{\Pi }}_N$ has decidable word problem.