George Kontogeorgiou

Discrete group actions on 3-manifolds and embeddable Cayley complexes

A classic theorem of Tucker asserts that a finite group Γ acts on an oriented surface S if and only if Γ has a Cayley graph G that embeds in S equivariantly, i.e. the canonical action of Γ on G can be extended to an action of Γ on all of S. Following the trend for extending graph-theoretic results to higher-dimensional complexes, we prove the following 3-dimensional analogue of Tucker’s Theorem: a finitely generated group Γ acts ​discretely on a simply connected 3-manifold M if and only if Γ has a “generalised Cayley complex” that embeds equivariantly in one of the following four 3-manifolds: (i) 𝕊³ , (ii) ℝ³, (iii) 𝕊² x ℝ, and (iv) the complement of a tame Cantor set in 𝕊³. In the process, we will see some recent theorems and lemmata concerning 2-complex embeddings and group actions over 2-complexes, and we will derive a combinatorial characterization of finitely generated groups acting ​discretely on simply connected 3-manifolds.