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Discrete group actions on 3-manifolds and embeddable Cayley complexes
- Part of
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 28 January 2025, pp. 1-26
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- Article
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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) 
$\mathbb {S}^2 \times \mathbb R$, and (iv) the complement of a tame Cantor set in 
$\mathbb {S}^3$. The fact that these are the only simply connected 3-manifolds that allow such actions is a consequence of the Thurston–Perelman geometrization theorem.
