Group cohomology, harmonic functions and the first \(L^ 2\)-Betti number
Let \(S\) be a finite symmetric generating subset of an infinite, finitely generated group \(\Gamma\) and let \(X={\mathcal G}(\Gamma,S)\) be the (unoriented) Cayley graph of \(\Gamma\) defined by \(S\), namely the graph with vertex set \(\Gamma\) and edge set the set of all pairs \(\{x,sx\}\) with \(x\in\Gamma\), \(s\in S\). The set of ends of \(\Gamma\) is the projective limit \(b(\Gamma)=\varprojlim_F\prod_0(X- F)\), where \(F\) runs over all finite subsets of \(\Gamma\) and \(\prod_0(X- F)\) is the set of all connected components of \(X-F\). The graph \(X\) becomes an oriented one by selecting a fundamental domain \(S^+\) for the action \(s\mapsto s^{-1}\) on \(S\), by defining the set \(E=\{(x,s^{-1}x)\mid x\in\Gamma\) and \(s\in S^+\}\) of the oriented edges of \(X\), and by putting \(e^+= s^{-1}x\), \(e^-= x\) for each oriented edge \(e= (x,s^{-1}x)\). Then the coboundary operator \(d\) of the space \({\mathcal F}(\Gamma)\) of all complex-valued functions on \(\Gamma\) into the space \({\mathcal F}(E)\) of all complex-valued functions on \(E\) is defined by \(df(e)=f(e^+)- f(e^-)\) \((e\in E)\), its formal adjoint \(d^*\) of \({\mathcal F}(E)\) into \({\mathcal F}(\Gamma)\) by \(d^*\xi(x)= \sum_{e:e^+=x}\xi(e^+)- \sum_{e:e^-=x}\xi(e^-)\) \((x\in\Gamma)\), and \(\Delta=d^*d\) is the Laplace operator on \({\mathcal F}(\Gamma)\). The space of harmonic functions with finite Dirichlet sum on \(\Gamma\) is \[ HD(X)= \{f|f\in{\mathcal F}(\Gamma)\text{ and }df\in\ell^2(E)\text{ and }\Delta f=0\} \] and \(HD(X)/\mathbb{C}\) is isomorphic to \[ H^1EL_2(X)= \{\xi|\xi\in \ell^2(E)\text{ and }d^*\xi= 0\text{ and }\xi\in d{\mathcal F}(\Gamma)\}, \] the first \(L^2\)-cohomology space of \(\Gamma\). The subject of this paper is the study of the first cohomology space \(H^1(\Gamma,\lambda_\Gamma)\), with coefficients in the space \(\ell^2(\Gamma)\) which carries the left regular representation \(\lambda_\Gamma\) of \(\Gamma\). It is proven that \(H^1(\Gamma,\mathbb{C}\Gamma)\) embeds into \(H^1(\Gamma,\lambda_\Gamma)\) and that \(\Gamma\) is nonamenable with one end when \(H^1(\Gamma,\lambda_\Gamma)= 0\). Moreover, if \(\Gamma\) is nonamenable, the spaces \(H^1(\Gamma,\lambda_\Gamma)\) and \(HD(X)/\mathbb{C}\) are isomorphic and this allows the use of graph-theoretical or \(L^2\)-cohomological results in order to obtain results on \(H^1(\Gamma,\lambda_\Gamma)\) or vice versa.
scientific article; zbMATH DE number 1045428