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On the submonoid membership problem for HNN extensions of free groups

Jonathan Warne

TL;DR

The paper addresses decidability of submonoid membership in HNN extensions of free groups and uses this to decide prefix membership and the word problem for certain one-relator inverse monoids. It develops a framework based on a restricted HNN form with bijective correspondences on a free basis, together with a finite set of elementary operations that preserves a form of freeness and yields algorithmic criteria for membership. These tools enable decidability results for a class of prefix membership problems, including explicit one-relator groups, and they apply to a concrete example previously unresolved in Dolinka–Gray. By connecting the Magnus–Moldavanskiĭ hierarchy with explicit combinatorial reductions, the work provides robust methods for studying word problems in inverse monoids arising from one-relator groups.

Abstract

We study membership problems in HNN extensions of free groups and then apply these results to solve the word problem in certain families of one-relator inverse monoids. In more detail, we consider HNN extensions where the defining isomorphism produces a bijection between subsets of a basis of the free group. Within such HNN extensions we identify natural conditions on submonoids of this group that suffice for membership in that submonoid to be decidable. We show that these results can then be applied to solve the prefix membership problem in certain one-relator groups which via results of Ivanov, Margolis and Meakin $(2001)$ then give solutions to the word problem for the corresponding one-relator inverse monoid. In particular our new techniques allow us to solve the word problem in an example (Example $7.6$) from Dolinka and Gray $(2021)$ which previous methods had not been able to resolve.

On the submonoid membership problem for HNN extensions of free groups

TL;DR

The paper addresses decidability of submonoid membership in HNN extensions of free groups and uses this to decide prefix membership and the word problem for certain one-relator inverse monoids. It develops a framework based on a restricted HNN form with bijective correspondences on a free basis, together with a finite set of elementary operations that preserves a form of freeness and yields algorithmic criteria for membership. These tools enable decidability results for a class of prefix membership problems, including explicit one-relator groups, and they apply to a concrete example previously unresolved in Dolinka–Gray. By connecting the Magnus–Moldavanskiĭ hierarchy with explicit combinatorial reductions, the work provides robust methods for studying word problems in inverse monoids arising from one-relator groups.

Abstract

We study membership problems in HNN extensions of free groups and then apply these results to solve the word problem in certain families of one-relator inverse monoids. In more detail, we consider HNN extensions where the defining isomorphism produces a bijection between subsets of a basis of the free group. Within such HNN extensions we identify natural conditions on submonoids of this group that suffice for membership in that submonoid to be decidable. We show that these results can then be applied to solve the prefix membership problem in certain one-relator groups which via results of Ivanov, Margolis and Meakin then give solutions to the word problem for the corresponding one-relator inverse monoid. In particular our new techniques allow us to solve the word problem in an example (Example ) from Dolinka and Gray which previous methods had not been able to resolve.

Paper Structure

This paper contains 9 sections, 23 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Words and Elements
  4. Factorisations
  5. HNN extensions
  6. Hierarchy
  7. Sequences of Elementary Operations
  8. Submonoids with decidable membership
  9. Application to the Prefix Membership Problem

Key Result

Theorem 1.1

Let $G = \gpPres[w = 1]$ be a group such that $w$ is $t$-sum zero, i.e. the amount of occurences of $t$ and $t^{-1}$ in $w$ are equal. Suppose that is a free group. If $w$ is $t$-prefix positive, i.e. every prefix of $w$ contains as least as many occurences of $t$ as occurences of $t^{-1}$, then $G$ has decidable prefix membership problem.

Figures (4)

  • Figure 1: The solid arrows represent cancellation at $m$, same as in the text. The dashed arrows denote the "cancellation" between the $w_i$ and their respective $v_i$. The particular sequence of $\equiv$ and $\sim$ between the $v_i$ is purely illustrative.
  • Figure 2: The squiggly lines represent a sequence of elementary semicommutations in the same way $\sim$ does in the body of the text.
  • Figure 3: The squiggly lines represent a sequence of elementary semicommutations in the same way $\sim$ does in the body of the text.
  • Figure 4: Double ended squiggly arrows indicate an indifference to direction. This demonstrates a case where the most extensive method is require.

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Example 2.6
  • Example 2.7
  • Lemma 3.1
  • proof
  • ...and 38 more