Homological Invariants and Quasi-Isometry

Abstract

Building upon work of Y. Shalom we give a homological-algebra flavored definition of an induction map in group homology associated to a topological coupling. As an application we obtain that the cohomological dimension cd R over a commutative ring R satisfies the inequality $$ \,{\text{cd}}_R (\Lambda ) \leq {\text{cd}}_R (\Gamma ) $$ if Λ embeds uniformly into Γ and $$ {\text{cd}}_R (\Lambda ) < \infty $$ holds. Another consequence of our results is that the Hirsch ranks of quasi-isometric solvable groups coincide. Further, it is shown that the real cohomology rings of quasi-isometric nilpotent groups are isomorphic as graded rings. On the analytic side, we apply the induction technique to Novikov-Shubin invariants of amenable groups, which can be seen as homological invariants, and show their invariance under quasi-isometry.