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We present a geometric method of proving that large scale homology groups of Cartesian products of non-compact spaces vanish and use it to obtain a number of vanishing result for several large scale homology theories in higher degrees. Our method applies for instance to Block and Weinberger's uniformly finite homology, Dranishnikov's almost equivariant homology and Nowak and \v{S}pakula's controlled coarse homology, where the standard K\"{u}nneth-type theorems are not available. As applications we determine group homology with $\ell_{\infty}$-coefficients of lattices in products of 2 and 3 trees, show a characterization of amenability in terms of 1-homology and construct aperiodic tilings using higher homology.

Eilenberg swindles and higher large scale homology of products of trees