Discrete group actions on 3-manifolds and embeddable Cayley complexes

TL;DR

This paper characterizes groups that can act discretely on simply-connected 3-manifolds by linking their properties to the embeddability of their Cayley complexes in specific 3-manifolds, providing a topological classification.

Contribution

It establishes a precise equivalence between group actions on 3-manifolds and embeddability of Cayley complexes in four particular 3-manifolds, advancing understanding of group actions in topology.

Findings

Groups act discretely on 3-manifolds iff their Cayley complexes embed in specific 3-manifolds.

Embeddability conditions are characterized by natural restrictions.

Provides a topological classification of groups based on their actions on 3-manifolds.

Abstract

We prove that a group $\Gamma$ admits a discrete topological (equivalently, smooth) action on some simply-connected 3-manifold if and only if $\Gamma$ has a Cayley complex embeddable -- with certain natural restrictions -- in one of the following four 3-manifolds: (i) $\mathbb{S}^3$, (ii) $\mathbb{R}^3$, (iii) $\mathbb{S}^2 \times \mathbb{R}$, (iv) the complement of a tame Cantor set in $\mathbb{S}^3$.