Rigidly achiral hyperbolic spatial graphs in 3-manifolds (Q2848022)

This paper provides a method for realising certain involutions of a graph as hyperbolic spatial embeddings of the graph in a 3-manifold. A spatial embedding of a graph \(G\) in a 3-manifold \(M\) is \textit{hyperbolic} if the exterior of the resulting spatial graph is hyperbolic, and is \textit{rigidly achiral} if there is an orientation-reversing periodic diffeomorphism of \(M\) that leaves the spatial graph setwise invariant. The main result of the paper is as follows. Let \(M\) be a compact connected orientable 3-manifold with non-empty aspherical boundary. Then there is some integer \(N\) such that when \(G\) is a graph with Euler characteristic less than \(N\) and minimum degree at least 2 with an involution that does not restrict to an orientation-preserving automorphism on any of its cycles, we have that the involution gives rise to a rigidly achiral hyperbolic spatial embedding of \(G\) into the double of \(M\) along \(\partial M\).