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.

Table of Contents:
Complex geometry in Chile: Panorama and perspectives by R. E. Rodriguez Quasiconformal conjugation of Mobius transformations by W. Abikoff A family of n-dimensional differential equations with Lorenz-like attractors by R. Bamon Automorphisms of $3$-dimensional abelian varieties by C. Birkenhake, H. Lange, and V. Gonzalez-Aguilera On restrictions of the Picard bundle by L. Brambila-Paz, L. Hidalgo-Solis, and J. Mucino-Raymundo Representations of triangle groups in the Heisenberg group by A. M. Castro Recent progress on the geometry of univalence criteria by M. Chuaqui and B. Osgood On anticonformal automorphisms of order $>2$ of Riemann by A. F. Costa and A. M. Porto Isomorphisms between generalized Teichmuller spaces by C. J. Earle, F. P. Gardiner, and N. Lakic Ramanujan partition identities by H. M. Farkas and I. Kra A function theoretic approach to the Ramanujan partition identities with applications to combinatorial number theory by H. M. Farkas and I. Kra Holomorphic motions and quasifuchsian manifolds by F. Gardiner and L. Keen Computation of topological numbers via linear algebra: hypersurfaces, vector fields and vector fields on hypersurfaces by L. Giraldo, X. Gomez-Mont, and P. Mardesic Cyclic coverings of a smooth curve and branch locus of the moduli space of smooth curves by E. Gomez Gonzalez Subvarieties of moduli space for Riemann surfaces by G. Gonzalez-Diez and W. J. Harvey Noded function groups by R. A. Hidalgo The hyperelliptic fibre of the Prym map in genus four by L. Hidalgo-Solis and S. Recillas-Pishmish From the shift loci to the connectedness loci of complex polynomials by J. Kiwi Some properties of the operator of exterior derivation on surfaces of revolution and $\mathbb{L_p}$-cohomology by Y. A. Kopylov Matrices for Fenchel-Nielsen parameters in genus $2$ by B. Maskit Arithmetic Fuchsian groups of signature $(0;e_1,e_2,e_3,e_4)$ with $2 \leq e_1 \leq e_2 \leq e_3$, $e_4 = \infty$ by T. Nakanishi, M. Naatanen, and G. Rosenberger Geometric and holomorphic moduli for extremal Riemann surfaces by J. R. Quine.