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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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