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

TL;DR

This paper constructs an infinite family of monoids with a single relation, demonstrating that their Dehn functions can grow exponentially, thus answering a longstanding open question and showing their word problems are decidable.

Contribution

It introduces a new class of monoids with exponential Dehn functions, providing a negative answer to a question about quadratic bounds for such monoids.

Findings

Dehn functions of the monoids grow at least exponentially

The monoids have decidable word problems

Answers a question about Dehn function bounds for monoids with one relation

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 $\partial_N(n) \succeq N^{n/4}$. 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 $n \geq 2$, proved recently by Cadilhac, Chistikov & Zetzsche, we show that each $\Pi_N$ has decidable word problem.