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The Cohen--Lyndon property in non-metric small-cancellation

Macarena Arenas, Karol Duda

TL;DR

The paper establishes the Cohen–Lyndon property for non-metric small-cancellation quotients of free groups, proving that for presentations $P=\langle S \mid R \rangle$ satisfying $C(6)$, $C(4)-T(4)$, or $C(3)-T(6)$, the normal closure $\langle\!\langle R\rangle\!\rangle$ has a basis given by conjugates of relators, i.e. $\langle\!\langle R\rangle\!\rangle = *_{i,t} \langle r_i\rangle^t$ via full left transversals. The authors introduce a topological framework, constructing a homotopy equivalence between a wedge of relator translates and the Cayley graph through an ordering on cycles and a key contraction lemma, yielding the main $C(6)$ result. The Appendix extends the approach to the $C(4)-T(4)$ and $C(3)-T(6)$ cases, with refined lemmas (Greendlinger-type, Helly-type) and ordering arguments to handle non-metric settings, including non-positively curved but not negatively curved groups. Together, these results generalize the metric $C'(\tfrac{1}{6})$ Cohen–Lyndon theory to a broader non-metric regime and resolve longstanding questions of Lyndon (1966) and Wall (1979).

Abstract

We show that the Cohen--Lyndon property holds for all non-metric small-cancellation quotients. This generalises the analogous result from the metric small-cancellation setting, and answers a question asked by Lyndon in 1966 and by Wall in his 1979 problem list.

The Cohen--Lyndon property in non-metric small-cancellation

TL;DR

The paper establishes the Cohen–Lyndon property for non-metric small-cancellation quotients of free groups, proving that for presentations satisfying , , or , the normal closure has a basis given by conjugates of relators, i.e. via full left transversals. The authors introduce a topological framework, constructing a homotopy equivalence between a wedge of relator translates and the Cayley graph through an ordering on cycles and a key contraction lemma, yielding the main result. The Appendix extends the approach to the and cases, with refined lemmas (Greendlinger-type, Helly-type) and ordering arguments to handle non-metric settings, including non-positively curved but not negatively curved groups. Together, these results generalize the metric Cohen–Lyndon theory to a broader non-metric regime and resolve longstanding questions of Lyndon (1966) and Wall (1979).

Abstract

We show that the Cohen--Lyndon property holds for all non-metric small-cancellation quotients. This generalises the analogous result from the metric small-cancellation setting, and answers a question asked by Lyndon in 1966 and by Wall in his 1979 problem list.
Paper Structure (8 sections, 28 theorems, 16 equations, 7 figures)

This paper contains 8 sections, 28 theorems, 16 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Structure and strategy:
  3. Background
  4. Small-cancellation notions
  5. Main Theorem
  6. Appendix: the C(4)-T(4) and C(3)-T(6) cases by Macarena Arenas and Karol Duda
  7. The C(4)-T(4) case
  8. The C(3)-T(6) case

Key Result

Theorem 1.1

Let $P=\langle s_1, \ldots, s_n \mid r_1, \ldots, r_k \rangle$ be a $C(6)$, $C(4)-T(4)$, or $C(3)-T(6)$ presentation, and let $N(\langle r_i\rangle)$ denote the normaliser of $\langle r_i\rangle$ in $\langle s_1, \ldots, s_n\rangle$. There exist full left transversals $T_i$ of $N(\langle r_i\rangle

Figures (7)

  • Figure 1: An annular diagram collaring a disc diagram in a $C(6)$ complex.
  • Figure 2: The ordering in Definition \ref{['def:order']} for a portion of a hexagonal grid.
  • Figure 3: Possible reductions in the last part of the proof of Lemma \ref{['clm:contractibleinduction']} when $\text{\sf Area} (D_\tau)=0$ and $D_\tau$ has branching.
  • Figure 4: Some annuli collaring disc diagrams in the example from Figure \ref{['fig:orderingex']}, and exhibiting the behaviour explained in Claim \ref{['clm:final']}.
  • Figure 5: The reductions used in the final part of the proof of Lemma \ref{['clm:contractibleinduction']} (replace $\beta$ with $\tau$ in the labelling) and in the inductive step of the proof of Claim \ref{['clm:final']}.
  • ...and 2 more figures

Theorems & Definitions (63)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1: Pieces
  • Definition 2.2: Disc diagram
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5: Annular diagrams and collared diagrams
  • Definition 2.6: Small-cancellation conditions
  • Definition 2.7: Ladders, shells, and spurs
  • Theorem 2.8: Greendlinger's Lemma
  • ...and 53 more