Homological Invariants and Quasi-Isometry

Details Homological Invariants and Quasi-Isometry Homological Invariants and Quasi-Isometry

2003-12-05

arxiv.org/abs/math/0312129v2

Mathematics

Algebraic Topology

33 pages; v2: major extension and change of v1 which contained only the result about Novikov-Shubin invariants

Scientific paper

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 estimates of the (co)homological dimension of groups G and H, where G embeds uniformly into H and the (co)homological dimension of G is finite. 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.

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