Eilenberg swindles and higher large scale homology of products of trees (Q2357415)

The uniformly fine homology \(H^{\mathrm{uf}}(X;R)\) was defined by \textit{J. Block} and \textit{S. Weinberger} [J. Am. Math. Soc. 5, No. 4, 907--918 (1992; Zbl 0780.53031)] and is a coarse homology theory for non-compact metric spaces, it is a quasi-isometric invariant and has many applications for coarse phenomena. The zero uniform homology has been used by Block and Weinberger [loc. cit.] to characterize amenability in spaces. On the other hand, let \(H^{(\infty )}_*\) denote the simplicial fine homology. The main theorem in this work is the following: Theorem 1. Let \(\Gamma_i, i=1, \ldots, n,\) be a family of bounded degree non-amenable graphs, let \(R\) be either \(\mathbb{Z}\) or \(\mathbb{R}\). Let \(X=\Gamma_1\times\dots \times \Gamma_n\) be their triangulated Cartesian product. Then, \(H^{(\infty )}_k (X;R)=H^{\mathrm{ae}}_k(X;R)=0\) for \(k=0,1,\ldots ,n-1\), where \(H^{\mathrm{\mathrm{ae}}}_*\) denotes \textit{A. Dranishnikov}'s almost equivariant homology [Geom. Topol. 15, No. 2, 1107--1124 (2011; Zbl 1220.53057)]. On the other hand, the authors prove that for a finite product of uniformly locally finite infinite trees, where vertices have degree at least 3, one has that their uniformly fine homology and simplicial fine homology coincide, thus giving the following theorem: Theorem 2. Let \(T_i\) be trees as before, \(i=1,\ldots ,n\) and let \(X=T_1\times\dots \times T_n\) be their Cartesian product. Then, \(H^{\mathrm{uf}}_n(X;R)\) is infinite dimensional and \(H^{\mathrm{uf}}_k(X;R)=0\) for \(k\not= n\), where \(R\) is as before.