A porism for cyclic quadrilaterals, butterfly theorems, and hyperbolic geometry

If there exists a cyclic quadrilateral whose sides go through the given four collinear points, then there are infinitely many such quadrilaterals inscribed in the same circle. We give two proofs of this porism; one based on cross-ratios, the other on compositions of hyperbolic isometries.Comment: 9 pages, 10 figures, to appear in Amer. Math. Monthl

Variational properties of the discrete Hilbert-Einstein functional

This is a survey on rigidity and geometrization results obtained with the help of the discrete Hilbert-Einstein functional, written for the proceedings of the "Discrete Curvature" colloquium in Luminy.Comment: 10 page

Branched Coverings, Triangulations, and 3-Manifolds

A canonical branched covering over each sufficiently good simplicial complex is constructed. Its structure depends on the combinatorial type of the complex. In this way, each closed orientable 3-manifold arises as a branched covering over the 3-sphere from some triangulation of S^3. This result is related to a theorem of Hilden and Montesinos. The branched coverings introduced admit a rich theory in which the group of projectivities plays a central role.Comment: v2: several changes to the text body; minor correction