Geometric & Functional Analysis GAFA

, Volume 16, Issue 2, pp 476–515 | Cite as

Homological Invariants and Quasi-Isometry

Original Paper

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.

Keywords and phrases.

Uniform embedding quasi-isometry nilpotent groups cohomological dimension Novikov–Shubin invariants 

2000 Mathematics Subject Classification.

Primary: 20F65 Secondary: 20F16 20F18 20F69 20J06 37A20 

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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.FB MathematikUniversität MünsterMünsterGermany

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