Analyzing the Sphere Complex of a Locally Finite Graph and its Mapping Class Group

The mapping class group of a locally finite graph Γ, denoted Map(Γ), acts faithfully on the sphere complex S(MΓ) of the associated doubled handlebody MΓ. This action is induced by a short exact sequence that splits topologically, giving an isomorphism Map(MΓ) ∼= Twists(MΓ) ⋊ Map(Γ), where Twists(MΓ) is a compact abelian group of sphere twists.