According to the classical uniformization theorem, every smooth Riemannian surface Z homeomorphic to the 2-sphere is conformally diffeomorphic to S 2 (the unit sphere

Following is a study of percolation in the hyperbolic plane H 2 and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and Poisson-Voronoi-Bernoulli percolation. We prove the existence of three distinct nonempty phases for the Bernoulli processes. In the first phase, p ∈ (0, pc], there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase, p ∈ (pc, pu), there are infinitely many unbounded clusters for the process and for the dual process. In the third phase, p∈[pu, 1), there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of pc in the Poisson-Voronoi-Bernoulli percolation process on the intensity of the underlying Poisson process.

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Abstract. A dimension reduction for hyperbolic space is established. When points are far apart, an embedding with bounded distortion into H 2 is achieved. 1.

We consider the Poisson-Boolean model with unit radius in the hyperbolic disc H 2. Let λ be the intensity of the underlying Poisson process, and let NC denote the number of unbounded components of the covered region. We show that there are two intensities λc and λu, 0 < λc < λu < ∞, such that NC = 0 for λ ∈ (0,λc], NC = ∞ for λ ∈ (λc,λu), and NC = 1 for λ ∈ [λu, ∞). Corresponding results, due to Benjamini, Lyons, Peres and Schramm, are available for Bernoulli bond and site percolation on certain nonamenable transitive graphs, and we use many of their techniques in our proofs. 1

Abstract. We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is one special tree reduced path. The set of these paths is the Reduced Path Group. It is a continuous analogue to the group of reduced words. The signature of the path is a power series whose coefficients are definite iterated integrals of the path. We identify the paths with trivial signature as the tree-like paths, and prove that two paths are in tree-like equivalence if and only if they have the same signature. In this way, we extend Chen’s theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for reparameterisation in the general setting. It is suggestive to think of this result as a non-commutative analogue of the result that integrable functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. As a second theme we give quantitative versions of Chen’s theorem in the case of lattice paths and paths with continuous derivative, and as a corollary derive results on the triviality of exponential products in the tensor algebra. 1.