On the Dehn Complex of Virtual Links

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A virtual link comes with a variety of link complements. This paper is concerned with the Dehn space, a pseudo-manifold with boundary, and the Dehn complex, a two-dimensional spine of the Dehn space. In the classical case where the link is planar, the Dehn space is the link complement in the 3-sphere. We study topological and geometric properties of the Dehn complex of a virtual link. Among other things, we show that every finitely presented group is the fundamental group of a Dehn complex, and that the Dehn complex of any alternating triple of an alternating virtual link is a non-positively curved squared complex.