In this thesis, we study the geography of complex surfaces of general type with respect to the topological fundamental group. The understanding of this general problem can be coarsely divided into geographyof simply-connected surfaces and geography of non-simply-connected surfaces. 

The geography of simply-connected surfaces was intensively studied in the eighties and nineties by Persson, Chen, and Xiao among others. Due to their works, we know that the set of Chern slopes c^2_1/c_2 of simply-connected surfaces of general type is dense in the interval [1/5,2]. The last result which closes the density problem for this type of surfaces happened in 2015. Roulleau and Urz\'ua showed the density of the Chern slopes in the interval [1,3]. This completes the study since accumulation points of c^2_1/c_2 belong to the interval [1/5,3] by the Noether's inequality and the Bogomolov-Miyaoka-Yau inequality for complex surfaces. 

The geography of non-simply-connected surfaces is well understood only for small Chern slopes. Indeed, because of works of Mendes, Pardini, Reid, and Xiao, we know that for c_1^2/c_2 in [1/5, 1/3] the fundamental group is either finite with at most nine elements, or the fundamental (algebraic) group is commensurable with the fundamental (algebraic) group of a curve. Furthermore, a well-known conjecture of Reid states that for minimal surfaces of general type with c_1^2/c_2< 1/2 the topological fundamental group is either finite or it is commensurable with the fundamental group of a curve. Due to  Severi-Pardini's inequality and a theorem of Xiao,  Reid's conjecture is true, at least in the algebraic sense for irregular surfaces or surfaces having an irregular étale cover. Keum showed with an example in his doctoral thesis that Reid's conjecture cannot be extended over 1/2. 

For higher slopes essentially there are no general results. In this thesis, we prove that for any topological fundamental group G of a given nonsingular complex projective surface, the Chern slopes c ^2_1(S)/c_2(S) of minimal nonsingular projective surfaces of general type S with pi_1(S) isomorphic to G are dense in the interval [1, 3]. It remains open the question for non-simply-connected surfaces in the interval [1/2,1]. 

Un invariante importante en el estudio de la geometría de un divisor es su dimensión de Iitaka. Ésta mide el crecimiento asintótico del espacio de secciones de los múltiplos del divisor. Sin embargo, la dimensión de Iitaka no se comporta bien en relación a la clase numérica del divisor, por lo que se han dado diversas definiciones que buscan capturar facetas de esta dimensión y que no dependan de la clase numérica.

En esta charla presentaré algunas definiciones de dimensión numérica de un divisor, junto con un ejemplo que evidencia problemas con estas definiciones y preguntas que surgen a partir de esto.