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The quadric flat torus theorem

Zachary Munro, Nima Hoda

Abstract

We prove a flat torus theorem for quadric complexes. In particular, we show that if a non-cyclic free abelian group $G$ acts metrically properly on a quadric complex $X$, then $G \cong \mathbb{Z}^2$ and $X$ contains a $G$-invariant isometric copy of the regular square tiling of the plane. Along the way, we also give a complete proof of the fact that any closed surface subgroup in the fundamental group of a combinatorial 2-complex is represented by a combinatorial map from a cellulation of the surface that is locally injective away from vertices.

The quadric flat torus theorem

Abstract

We prove a flat torus theorem for quadric complexes. In particular, we show that if a non-cyclic free abelian group acts metrically properly on a quadric complex , then and contains a -invariant isometric copy of the regular square tiling of the plane. Along the way, we also give a complete proof of the fact that any closed surface subgroup in the fundamental group of a combinatorial 2-complex is represented by a combinatorial map from a cellulation of the surface that is locally injective away from vertices.

Paper Structure

This paper contains 9 sections, 28 theorems, 1 equation, 8 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Quadric Complexes
  4. Short cycles in quadric complexes
  5. Diagrams in quadric complexes
  6. Quadric geodesics
  7. Isometric Flats
  8. Thick Flats
  9. The Flat Torus Theorem

Key Result

Theorem I

Let $G$ be a non-cyclic free abelian group. Let $G$ act metrically properly on a quadric complex $X$ (e.g. let $G$ act freely on a locally finite quadric complex $X$). Then $G \cong \mathbb{Z}^2$ and there is a $G$-invariant flat in $X$.

Figures (8)

  • Figure 1: The fold map $f$. Let $g \colon s_1 \to s_2$ be an isomorphism of combinatorial complexes where $s_1$ and $s_2$ are squares. Let $P_1 \subset \partial s_1$ be a combinatorial path of length $3$. The domain of the fold map is $s_1 \sqcup s_2 /{\sim}$ where $x \sim g(x)$ for $x \in P_1$. The fold map is the quotient of $s_1 \sqcup s_2 /{\sim}$ identifying $[x]_{\sim}$ and $[g(x)]_{\sim}$ for all $x \in s_1$.
  • Figure 2: Replacement rules for quadric complexes.
  • Figure 3: A $6$-cycle to which we apply Lemma \ref{['lem:sixcycles']}.
  • Figure 4: Factoring through a $6$-cycle wedge an edge.
  • Figure 5: Factoring through a wedge of two $4$-cycles.
  • ...and 3 more figures

Theorems & Definitions (61)

  • Theorem I: The Quadric Flat Torus Theorem
  • Theorem II
  • Theorem III
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5: van Kampen's Lemma
  • Definition 2.6
  • Lemma 2.7: Combinatorial Gauss-Bonnet
  • ...and 51 more