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Harmonic analysis, cohomology, and the large-scale geometry of amenable groups. (English) Zbl 1064.43004
This paper explores aspects of geometric group theory involving representation theory and cohomology. Two groups \(\Gamma\) and \(\Lambda\) generated by finite symmetric sets are called quasi-isomorphic if there is a map \(\phi\) from \(\Lambda\) to \(\Gamma\) and constants \(\alpha\geq 1\) and \(K\geq 0\) such that for all \(\lambda_1\), \(\lambda_2\) in \(\Lambda\) \[ \alpha^{-1} d_\Lambda(\lambda_1, \lambda_2)- K\leq d_\Gamma(\phi\lambda_1, \phi\lambda_2)\leq \alpha d_\Lambda(\lambda_1, \lambda_2)+ K \] and any element of \(\Lambda\) is within distance \(\leq K\) from \(\phi\Lambda\). Here \(d_\Lambda\) and \(d_\Gamma\) are the word metrics on the respective groups. The author explores various implications of quasi-isometry to a host of settings.
The first result is that if \(\Gamma\) is quasi-isomorphic to \(\mathbb{Z}^d\), then \(\Gamma\) has a finite index subgroup isomorphic to \(\mathbb{Z}^d\). Other results that bring in cohomology for quasi-isomorphic groups are as follows: If \(\Gamma\) and \(\Lambda\) are finitely generated, then all of their respective Betti numbers are equal. If \(\Gamma\) is quasi-isomorphic to a polycyclic group, then all of its virtual Betti numbers are greater than zero. The cohomological dimensions of \(\Gamma\) and \(\Lambda\) over a ring \(R\) are identical. The author also applies his techniques to a class of non-finitely presentable abelian-by-cyclic groups.
The author discusses a notion more general than quasi-isometry, namely, uniform embedding. Many of the results for quasi-isomorphic groups hold for uniformly embedded pairs of groups. For example, cohomological dimensions are preserved. Having the type (FP) property and for solvable groups \(\Gamma\) and \(\Lambda\), the Hirsh number of \(\Lambda\) is bounded by that of \(\Gamma\).
Let \(\Gamma\) be a discrete group. Definition: \(\Gamma\) has property \(H_{FD}\) if for every unitary representation \(\pi\) of \(\Gamma\) with non-zero first cohomology with respect to \(\pi\) there exists a finite-dimensional \(\Gamma\)-subrepresentation. The author proves that among amenable groups, the property \(H_{FD}\) is a quasi-isometry invariant.

MSC:
43A07 Means on groups, semigroups, etc.; amenable groups
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
20J06 Cohomology of groups
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