Homological Invariants and Quasi-Isometry

By Roman Sauer. 2003

76 citations recorded by the source corpus.

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 cdR 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6340d4c6eebe40f10923593a5630bf3e7847d4ad

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In Geometry and Topology

1. Gregory C. Bell
2. Alexander Dranishnikov
3. D. Fisher
4. A. Eskin
5. K. Whyte
6. A. Furman
7. Xin Li
8. Mark F. Hagen

In Mathematical Physics

1. Gregory C. Bell
2. Alexander Dranishnikov
3. K. Whyte
4. A. Furman
5. D. Fisher
6. A. Eskin
7. Xin Li
8. Mark F. Hagen

Citations in the corpus

• Asymptotic Dimension (2007, background): “This result was extended to solvable groups [92],[86].”
• A SURVEY OF MEASURED GROUP THEORY (2009, background): “The view point of topological dynamics of the Γ-action on [Γ,Λ] may be related to quasi-isometries and Topological Equivalence.”
• Continuous orbit equivalence rigidity (2015, background): “w present explicit constructions of continuous orbit couples out of topological couplings and vice versa. The constructions are really the topological analogues of those in [7, §3] (see also [23] and [20]). 6.1. From topological couplings to continuous orbit couples. Suppose that we are given a topo”
• Quasi-isometries and rigidity of solvable groups (2005, background): “Figure 1. The trees for DL(3;2). Figure borrowed from [ PPS ].”