Can every abstract simplicial complex whose geometric realization is homeomorphic to $\mathbb{R}^2$ be realized by a rectilinear triangulation of the Euclidean plane? Alternatively put, can a curvy (honest) triangulation of $\mathbb{R}^2$ be straightened?

In a related question (Euclidean triangulation of the plane with degree 7 at each vertex.) the answers show in effect that a geodesic triangulation of the hyperbolic plane is combinatorially equivalent to a rectilinear triangulation of the Euclidean plane. It would therefore suffice to realize an abstract triangulation in the hyperbolic plane.

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    $\begingroup$ (A contrast to your question is that there exist "unstretchable" pseudoline arrangements, curvy arrangements that cannot be realized in the plane by straight lines.) $\endgroup$ – Joseph O'Rourke Feb 20 '15 at 15:54

Yes. For finite complexes this is completely standard, for infinite graphs, this follows by a compactness argument (in particular, there is an infinite circle packing, though it is not unique in this case). You can see my old paper Rivin, Igor. "Combinatorial optimization in geometry." Advances in Applied Mathematics 31.1 (2003): 242-271., or one of Z. He and O. Schramm's paper (for example: He, Zheng-Xu, and Oded Schramm. "Hyperbolic and parabolic packings." Discrete & Computational Geometry 14.1 (1995): 123-149.)


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