Cellular Waists of Hyperbolic Spaces

We find lower bounds on the topological complexity of fibers of PL and generic smooth maps p:Mᵈ→ Rᵐ p : M d → R m , where Mᵈ M d is a closed hyperbolic manifold of large injectivity radius. More precisely, we show that if the injectivity radius of M M is greater than 50log ((n+1)!) 50 log ( ( n + 1 ) ! ) , then for each dimension $0< k< d-m$ 0 < k < d − m there is a point z∈ Rᵐ z ∈ R m such that any cell structure on the fiber p⁻¹(z) p − 1 ( z ) has more than n n cells of dimension k k . The proof is based on the freedom theorem for ideals in group rings of hyperbolic groups proved in [AD26].