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Discrete group actions on 3-manifolds and embeddable Cayley complexes

Agelos Georgakopoulos, George Kontogeorgiou

TL;DR

The paper develops a complete bridge between discrete group actions on simply-connected 3-manifolds and embeddable Cayley-type 2-complexes. It introduces invariant planar rotation systems and uses Carmesin’s embedding criteria, together with geometrization (Agol) and smoothing results (Pardon), to characterize when a finitely generated group acts discretely and cocompactly on a simply-connected 3-manifold via an embeddable Cayley complex in one of four model spaces: $\mathbb{S}^3$, $\mathbb{R}^3$, $\mathbb{S}^2\times\mathbb{R}$, or the Cantor 3-sphere. The authors provide two routes: (i) from invariant Cayley complex embeddings to group actions using a Whitney-type extension and contraction techniques to realize the action on vertices, and (ii) from group actions to embedded invariant Cayley complexes by constructing finite-chamber models and refining them to vertex-regular generalized Cayley complexes via a detailed ‘fruit’ construction. The results unify and extend classic 2D planar-group correspondences (Maschke-type) to 3D, yielding a robust framework linking combinatorial, geometric, and topological aspects of discrete group actions on 3-manifolds with tangible embedding data. These contributions have implications for understanding which groups can act discretely on 3-manifolds and how their Cayley complexes can be realized geometrically, with potential applications to study of Kleinian groups and 3-manifold topology.

Abstract

We prove that a group $Γ$ admits a discrete topological (equivalently, smooth) action on some simply-connected 3-manifold if and only if $Γ$ 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 $\mathbb{S}^3$.

Discrete group actions on 3-manifolds and embeddable Cayley complexes

TL;DR

The paper develops a complete bridge between discrete group actions on simply-connected 3-manifolds and embeddable Cayley-type 2-complexes. It introduces invariant planar rotation systems and uses Carmesin’s embedding criteria, together with geometrization (Agol) and smoothing results (Pardon), to characterize when a finitely generated group acts discretely and cocompactly on a simply-connected 3-manifold via an embeddable Cayley complex in one of four model spaces: , , , or the Cantor 3-sphere. The authors provide two routes: (i) from invariant Cayley complex embeddings to group actions using a Whitney-type extension and contraction techniques to realize the action on vertices, and (ii) from group actions to embedded invariant Cayley complexes by constructing finite-chamber models and refining them to vertex-regular generalized Cayley complexes via a detailed ‘fruit’ construction. The results unify and extend classic 2D planar-group correspondences (Maschke-type) to 3D, yielding a robust framework linking combinatorial, geometric, and topological aspects of discrete group actions on 3-manifolds with tangible embedding data. These contributions have implications for understanding which groups can act discretely on 3-manifolds and how their Cayley complexes can be realized geometrically, with potential applications to study of Kleinian groups and 3-manifold topology.

Abstract

We prove that a group admits a discrete topological (equivalently, smooth) action on some simply-connected 3-manifold if and only if has a Cayley complex embeddable -- with certain natural restrictions -- in one of the following four 3-manifolds: (i) , (ii) , (iii) , (iv) the complement of a tame Cantor set in .
Paper Structure (18 sections, 27 theorems, 1 equation, 3 figures)

This paper contains 18 sections, 27 theorems, 1 equation, 3 figures.

Table of Contents

  1. Introduction
  2. Finite group actions on $\mathbb S^2$ and $\mathbb S^3$
  3. Infinite discrete actions on open 3-manifolds
  4. Definitions and preliminary results
  5. Group actions
  6. Graphs
  7. 2-complexes
  8. Embeddings and Chambers
  9. Local flatness
  10. Rotation systems
  11. Invariant rotation systems
  12. From Invariant Planar Rotation Systems to Invariant Cayley Complex Embeddings
  13. From Invariant Cayley Complex Embeddings to Group Actions
  14. From Group Actions to Invariant Embedded generalised Cayley complex es
  15. Step 1: Constructing an embedded $2$-complex with a regular action on its chambers
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $M$ be a connected, simply connected, topological 3-manifold. Suppose $M$ admits a properly-discontinuous, co-compact action by homeomorphisms. Then $M$ is homeomorphic to one of the following four spaces: $\mathbb S^3$, or $\mathbb R^3$, or $\mathbb S^2 \times \mathbb R$, or the Cantor 3-sphere

Figures (3)

  • Figure 1: A portion of a graph $G$ (left half), and the corresponding part of $G^\otimes$ (right half).
  • Figure 2: An example $F(X)$, when $X$ is the cubic lattice (top left). Each link graph is an octahedron (top right), and so each pineapple of $F(X)$ is a truncated cuboctahedron (bottom left). The pineapples are arranged as in the bottom right figure, which shows four of them in the front.
  • Figure 3: A topological 5-gon with a slice pattern. The $E_i$ are depicted in red (if colour is shown), and the $V_i$ in blue.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2: Folklore
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1: Whitney3D
  • Theorem 3.1: CarEmbII
  • Remark 1
  • Theorem 3.2
  • proof
  • ...and 39 more