The work provides partial answers (cf. Theorem 1.5) to the following conjecture: For any symmetric space of noncompact type \(S\), there are constants \(\alpha(S), d(S)\), such that any irreducible \(S\)-manifold \(\,M=\Gamma\setminus S\,\) (which is assumed also to be arithmetic in the case \(\,\text{dim}S=3\,\)) is homotopically equivalent to a simplicial complex with at most \(\alpha(S)\,\text{vol}(M)\) vertices, all of them of valence at most \(d(S)\). For example, the conjecture holds for noncompact arithmetic \(S\)-manifolds. The results imply quantitative versions for some classical finiteness statements. The author provides a linear bound on the size of a minimal presentation of the fundamental group in terms of the volume and is able to estimate the number of \(S\)-manifolds with bounded volume.
The basic idea behind the main result of Theorem 1.5 is to construct, inside each \(S\)-manifold \(M\), a submanifold \(M'\), which is similar enough to \(M\) and for which one can construct a triangulation of size \(\,c\cdot\text{vol}(M)\,\). For this construction, one requires a lower bound on the injectivity radius of \(M'\) (independent of \(M\)) and some bounds on the geometry of the boundary of \(M'\).