Imagine you have a donut-shaped or pretzel-shaped space that can be given a perfectly uniform, negatively curved geometry, like a saddle that curves the same way in every direction everywhere. Such spaces are called hyperbolic manifolds, and hyperbolic geometry is in many ways the "optimal" or most symmetric way to put a geometry on these spaces. Now suppose someone comes along and deforms the geometry, bending and stretching it in non-uniform ways. The question this paper addresses is: can we still say something meaningful about the geometry of the deformed space, particularly about how large balls grow as you increase their radius?
The paper proves a comparison theorem about ball volumes on the universal cover, which is the infinite "unrolled" version of the manifold where all the loops have been unwound. Specifically, it shows that if you put any Riemannian metric (a general, possibly non-uniform curved geometry) on a hyperbolic manifold, and if the total volume of the manifold is small relative to a topological quantity called the simplicial volume (which measures the topological complexity of the space using triangles), then balls of radius at least 1 in the deformed geometry are at least as large as the corresponding balls in the standard hyperbolic space. In other words, even under significant deformation, the geometry cannot be "too compressed" once the volume efficiency condition is met.
This matters because it connects topology to geometry in a precise and quantitative way. The simplicial volume is a purely topological invariant, meaning it does not depend on the geometry at all, and the result says that topological complexity forces geometric largeness. This fits into a broader program in geometric topology of understanding which geometric properties are rigid, meaning they persist even when you deform the space, and which are flexible. Results like this have implications for understanding the spectrum of possible geometries on a given topological space and for proving rigidity theorems that show hyperbolic geometry is in some sense extremal or optimal.