Let $G$ be a group. We say that $G$ is coherent if every finitely generated subgroup of $G$ is finitely presented. One can view coherence as being like a compactness condition. It is known that a covering of a compact surface is another compact surface or a surface with free fundamental group. However, every finitely presented group is the fundamental group of some compact 4-manifold; and yet there are incoherent finitely presented groups. The coherence of groups is a topic of great interest because it is generally more manageable to probe questions about the structure of a group that can be described using a finite amount of data. There has been a lot of work to detect coherence in specific families of groups. A major discovery was that of Peter Scott in his 1973 paper, titled "Finitely generated 3-manifold groups are finitely presented," showing that any finitely generated group that is the fundamental group of a 3-manifold is finitely presented, which is interesting because we do not have a simple classification of the fundamental groups of 3-manifolds.
From a topological point of view, finitely generated groups are fundamental groups of cell complexes with a finite 1-skeleton; i.e., a graph. Finitely presented groups are the fundamental groups of cell complexes with finite 2-skeleton. If one takes a (potentially infinite sheeted) covering space that happens to have a finitely generated fundamental group, there is no reason this cell complex has its fundamental group supported on a finite subcomplex.
Some of the tools used to attack this problem are developed in a wonderful 1983 paper by John Stallings, titled "Topology of finite graphs." I hope to share some of the tools that are developed in this paper. Then, I will mention a 2017 paper by Larsen Louder and Henry Wilton, titled "Stackings and the W-cycles conjecture," which develops the technology of stackings to make progress towards answering the question of whether every one-relator group is coherent. In this talk I assume familiarity with the basics of algebraic topology, say to the level of the first chapter of Hatcher.