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This volume presents the proceedings of the I Iberoamerican Congress on Geometry: Cruz del Sur held in Olmue, Chile. The main topic was 'The Geometry of Groups: Curves, Abelian Varieties, Theoretical and Computational Aspects'. Participants came from all over the world. The volume gathers the expanded contributions from most of the participants in the Congress. Articles reflect the topic in its diversity and unity, and in particular, the work done on the subject by Iberoamerican mathematicians. Original results and surveys are included on the following areas: curves and Riemann surfaces, abelian varieties, and complex dynamics. The approaches are varied, including Kleinian groups, quasiconformal mappings and Teichmuller spaces, function theory, moduli spaces, automorphism groups, algebraic geometry, and more.