The small cancellation flat torus theorem

TL;DR

This paper extends flat torus theorems to small cancellation complexes under various conditions, revealing new phenomena and introducing concepts like thickened-flats and quasi-flats.

Contribution

It establishes flat torus results for small cancellation complexes, introduces thickened-flats, and demonstrates the existence and non-existence of flats in different settings.

Findings

C(3)-T(6) complexes closely parallel CAT(0) spaces.

Explicit example of a C(4)-T(4) complex with no invariant flat.

Flat torus theorem extended to quasi-flats via quadrization.

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 invariant under automorpihsms. We introduce the notion of thickened-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.