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The small cancellation flat torus theorem

Karol Duda

TL;DR

This work extends Flat Torus-type results to small cancellation complexes under $C(6)$, $C(4)$--$T(4)$, and $C(3)$--$T(6)$. It proves a CAT$(0)$-style invariant flat for $C(3)$--$T(6)$ complexes, and a corresponding flat in the systolic dual for $C(6)$ complexes, by transferring flats between the original complex and its nerve or dual. In the $C(4)$--$T(4)$ case, genuine flats may fail to exist; nevertheless, a Flat Torus theorem for quasi-flats is obtained through quadrization and the Quadric Flat Torus Theorem, and an explicit counterexample shows the limitations of CAT$(0)$ approaches in this setting. Overall, the paper connects small cancellation theory with geometric techniques, clarifying when invariant flats (or quasi-flats) exist and how dualities (systolic/quadric) mediate these phenomena.

Abstract

We establish Flat Torus Theorem type results for groups acting on small cancellation complexes satisfying C(6), C(4)-T(4) and C(3)-T(6) conditions. For C(3)-T(6) complexes the result closely parallels the CAT(0) setting. For C(6) complexes we prove an analogous theorem using a refined notion of flat, exploiting the relationship between C(6) complexes and their duals. In the C(4)-T(4) case we demonstrate that genuine flats do not necessarily exist, providing an explicit example of a C(4)-T(4) complex with an action of $\mathbb{Z}^2$ without invariant flat, and hence not admitting any CAT(0) metric. We introduce the notion of quasi-flats and prove a Flat Torus Theorem for quasi-flats by passing to quadric complexes via quadrization and invoking the Quadric Flat Torus Theorem of Hoda-Munro.

The small cancellation flat torus theorem

TL;DR

This work extends Flat Torus-type results to small cancellation complexes under , --, and --. It proves a CAT-style invariant flat for -- complexes, and a corresponding flat in the systolic dual for complexes, by transferring flats between the original complex and its nerve or dual. In the -- case, genuine flats may fail to exist; nevertheless, a Flat Torus theorem for quasi-flats is obtained through quadrization and the Quadric Flat Torus Theorem, and an explicit counterexample shows the limitations of CAT approaches in this setting. Overall, the paper connects small cancellation theory with geometric techniques, clarifying when invariant flats (or quasi-flats) exist and how dualities (systolic/quadric) mediate these phenomena.

Abstract

We establish Flat Torus Theorem type results for groups acting on small cancellation complexes satisfying C(6), C(4)-T(4) and C(3)-T(6) conditions. For C(3)-T(6) complexes the result closely parallels the CAT(0) setting. For C(6) complexes we prove an analogous theorem using a refined notion of flat, exploiting the relationship between C(6) complexes and their duals. In the C(4)-T(4) case we demonstrate that genuine flats do not necessarily exist, providing an explicit example of a C(4)-T(4) complex with an action of without invariant flat, and hence not admitting any CAT(0) metric. We introduce the notion of quasi-flats and prove a Flat Torus Theorem for quasi-flats by passing to quadric complexes via quadrization and invoking the Quadric Flat Torus Theorem of Hoda-Munro.
Paper Structure (11 sections, 20 theorems, 12 equations, 9 figures)

This paper contains 11 sections, 20 theorems, 12 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Small cancellation
  4. Flats
  5. Proof of Theorem \ref{['flattorus36']}
  6. Case of $C(6)$ complexes
  7. Systolic complexes and nerve complex
  8. Flat torus
  9. The case of $C(4)$--$T(4)$ complexes
  10. Quadric complexes and quadrization
  11. Flats in $C(4)$--$T(4)$ complexes

Key Result

Theorem 1

Let $G$ be a free-abelian group of rank $n$ acting metrically properly and semisimply on a CAT(0) space $\widetilde{X}$. There exists a subspace $V\times F \subset \widetilde{X}$ with $F$ isometric to $\mathbb{E}^n$ such that $G$ stabilizes $V\times F$ and acts as: $g(v,f)=(v,gf)$ for all $(v,f)\i

Figures (9)

  • Figure 1: Condition (C) and (D) of Definition 7.1
  • Figure 2:
  • Figure 3:
  • Figure 4: Complex $\mathfrak{X}'$ from Definition \ref{['quasi-flat']} with single $2$-cell marked.
  • Figure 5: Example of numberings $\phi$ and $\psi$ of $X$ and $\mathfrak{X}'$
  • ...and 4 more figures

Theorems & Definitions (42)

  • Theorem 1: Flat torus theorem
  • Theorem 2: Flat Torus for $C(3)$--$T(6)$ complexes
  • Theorem 3: Flat Torus for $C(6)$ complexes
  • Theorem 4: Flat Torus for $C(4)$--$T(4)$ complexes
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • Definition 3.4
  • Definition 3.5
  • Lemma 3.6
  • ...and 32 more