Cusps are Dense

Cusps Are Dense

We show cusps are dense in Bers' boundary for Teichm uller space. The proof rests on an estimate for the algebraic eeect of a unit quasiconformal deformation supported in the thin part of a hyperbolic Riemann surface.

Dense graphs are antimagic

An antimagic labeling of a graph with m edges and n vertices is a bijection from the set of edges to the integers 1, . . . ,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. A conjecture of Ringel (see [4]) states that every connected graph, but K2,...

Dynamical Dessins Are Dense

We apply a recent result of the first author to prove the following result: any continuum in the plane can be approximated arbitrarily closely in the Hausdorff topology by the Julia set of a critically finite polynomial with two finite postcritical points.

True Trees Are Dense

We show that any compact, connected set K in the plane can be approximated by the critical points of a polynomial with two critical values. Equivalently, K can be approximated in the Hausdorff metric by a true tree in the sense of Grothendieck’s dessins d’enfants. Mathematics Subject Classification 30C62

All at Manifolds Are Cusps of Hyperbolic Orbifolds