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Every non-trivial knot group is fully residually perfect

Tetsuya Ito, Kimihiko Motegi, Masakazu Teragaito

TL;DR

The paper addresses whether every non-trivial knot group $G(K)$ is fully residually perfect, aiming to realize finite subsets of nontrivial elements via epimorphisms to perfect (closed 3-manifold) targets. It develops a Dehn-filling framework combined with a graph-of-groups decomposition of $G(K)$ along its torus JSJ pieces, and introduces a precise slope-analysis via conditions like $(g|X)$ to guarantee nontrivial survivals of elements after fillings. For prime non-cabled knots, it proves that each fixed nontrivial element can be prevented from vanishing under all but finitely many Dehn fillings, and extends the argument to all nontrivial knots via $1/n$-fillings, concluding that every nontrivial knot group is fully residually perfect, closed $3$-manifold group. The results unify hyperbolic, torus, and satellite knot cases, and yield a structural perspective on residual properties of knot groups with implications for the algebraic structure of 3-manifold groups. The approach also clarifies limitations: while hyperbolic knot groups can be fully residually closed hyperbolic 3-manifold groups, certain satellite configurations preclude residual-closedness in that stronger sense, though full residually perfect behavior still holds via $1/n$-type fillings.

Abstract

Given a class $\mathcal{P}$ of groups we say that a group $G$ is fully residually $\mathcal{P}$ if for any finite subset $F$ of $G$, there exists an epimorphism from $G$ to a group in $\mathcal{P}$ which is injective on $F$. It is known that any non-trivial knot group is fully residually finite. For hyperbolic knots, its knot group is fully residually closed hyperbolic $3$--manifold group, and fully residually simple. In this article, we show that every non-trivial knot group is fully residually perfect, closed $3$--manifold group.

Every non-trivial knot group is fully residually perfect

TL;DR

The paper addresses whether every non-trivial knot group is fully residually perfect, aiming to realize finite subsets of nontrivial elements via epimorphisms to perfect (closed 3-manifold) targets. It develops a Dehn-filling framework combined with a graph-of-groups decomposition of along its torus JSJ pieces, and introduces a precise slope-analysis via conditions like to guarantee nontrivial survivals of elements after fillings. For prime non-cabled knots, it proves that each fixed nontrivial element can be prevented from vanishing under all but finitely many Dehn fillings, and extends the argument to all nontrivial knots via -fillings, concluding that every nontrivial knot group is fully residually perfect, closed -manifold group. The results unify hyperbolic, torus, and satellite knot cases, and yield a structural perspective on residual properties of knot groups with implications for the algebraic structure of 3-manifold groups. The approach also clarifies limitations: while hyperbolic knot groups can be fully residually closed hyperbolic 3-manifold groups, certain satellite configurations preclude residual-closedness in that stronger sense, though full residually perfect behavior still holds via -type fillings.

Abstract

Given a class of groups we say that a group is fully residually if for any finite subset of , there exists an epimorphism from to a group in which is injective on . It is known that any non-trivial knot group is fully residually finite. For hyperbolic knots, its knot group is fully residually closed hyperbolic --manifold group, and fully residually simple. In this article, we show that every non-trivial knot group is fully residually perfect, closed --manifold group.

Paper Structure

This paper contains 18 sections, 19 theorems, 27 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Fully residual property
  3. Decomposing the knot group and its elements
  4. Convention on the fundamental groups
  5. Decomposition of elements of $G(K)$
  6. Dehn filling trivialization
  7. Dehn fillings of a hyperbolic manifold
  8. Dehn fillings of a composing space
  9. Dehn fillings of a cable space
  10. Dehn filling of a torus knot space
  11. Fundamental group of graph of groups
  12. Before Dehn fillings
  13. After Dehn fillings
  14. Condition for a non-trivial element surviving after Dehn fillings
  15. Proofs of Theorems \ref{['prime_non-cable']} and \ref{['1/n']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $K$ be a hyperbolic knot. Then its knot group is fully residually closed hyperbolic $3$--manifold group.

Figures (7)

  • Figure 3.1: The subgroups $\pi_1(E_i)$
  • Figure 3.2: Decomposition of elements as products of elements in factor groups
  • Figure 5.1: Graph of groups
  • Figure 5.2:
  • Figure 5.3: backtrack condition: $x_i \not\in G_{y_i}^{t(y_i)} \subset G_{t(y_i)}$ for a backtrack $y_i x_i y_{i+1} = y_i x_i \overline{y_i}$
  • ...and 2 more figures

Theorems & Definitions (38)

  • Theorem 1.1: AFW
  • Theorem 1.2: GMOsinIchiharaMT
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Proposition 2.1
  • proof
  • Lemma 2.2: IMT_Magnus
  • ...and 28 more