The quadric flat torus theorem

TL;DR

This paper establishes a flat torus theorem for quadric complexes, showing that certain free abelian groups acting on these complexes contain invariant square tilings, and provides a proof regarding surface subgroups in 2-complexes.

Contribution

It introduces a flat torus theorem specific to quadric complexes and offers a complete proof about surface subgroups in combinatorial 2-complexes.

Findings

Non-cyclic free abelian groups acting on quadric complexes are isomorphic to Z^2.

Such complexes contain invariant isometric square tilings of the plane.

A complete proof that surface subgroups in 2-complex groups are represented by locally injective maps.

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.