The three-dimensional homology cobordism group is a fundamental object in low-dimensional topology. A major challenge in the study of its structure is to understand the interaction between hyperbolic geometry and homology cobordism. In this talk, after introducing the main protagonists and explain the role they play in topology, I will discuss how monopole Floer homology can be used to study some basic properties of certain subgroups of the homology cobordism group generated by hyperbolic homology spheres satisfying natural geometric constraints of Riemannian and spectral nature.