Title: Quantitative bordism over acyclic groups and Cheeger-Gromov $ρ$-invariants

Abstract: We prove a bordism version of Gromov's linearity conjecture over a large family of acyclic groups. Since all groups embed into these acyclic groups, it follows that the linear bordism conjecture is true if one allows to enlarge a given group. Our result holds in both PL and smooth categories, and for both oriented and unoriented cases. The method of the proof hinges on quantitative algebraic and geometric techniques over infinite complexes with unbounded local (combinatorial) geometry, which seem interesting on their own. As an application, we prove that there is a universal linear bound for the Cheeger-Gromov $L^2$ $\rho$-invariants of PL $(4k-1)$-manifolds associated with arbitrary regular covers.