Let PU (2, 1) be the group of holomorphic isometries in the hyperbolic complex plane H 2 C and let G n be a sub-group of PU (2, 1) which is generated by n complex reflections with respect to complex lines in H 2 C . Under certain conditions, we prove that G n is discrete. We construct representations ρ of the fundamental group g of the compact surface g of genus g, into PU (2, 1), we prove they are discrete, faithful and we compute the dimension their deformation space.

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The well-known Broken Spaghetti Problem is a geometric problem which can be stated as: A stick of spaghetti breaks into three parts and all points of the stick have the same probability to be a breaking point. What is the probability that the three sticks, putting together, form a triangle? In this note, we describe a hidden geometric pattern behind the symmetric version of this problem, namely a fractal that parametrizes the sample space of this problem. Using that fractal, we address the question about the probability to obtain a δ-equilateral triangle.

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