  A major challenge in the study of the structure of the three-dimensional
homology cobordism group is to understand the interaction between hyperbolic
geometry and homology cobordism. In this paper, for a hyperbolic homology
sphere $Y$ we derive explicit bounds on the relative grading between
irreducible solutions to the Seiberg-Witten equations and the reducible one in
terms of the spectral and Riemannian geometry of $Y$. Using this, we provide
explicit bounds on some numerical invariants arising in monopole Floer homology
(and its $\mathrm{Pin}(2)$-equivariant refinement). We apply this to study the
subgroups of the homology cobordism group generated by hyperbolic homology
spheres satisfying certain natural geometric constraints.
