Groups, Geometry, and Dynamics


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Volume 11, Issue 1, 2017, pp. 371–392
DOI: 10.4171/GGD/400

Published online: 2017-04-20

Eilenberg swindles and higher large scale homology of products of trees

Francesca Diana[1] and Piotr W. Nowak[2]

(1) Universit├Ąt Regensburg, Germany
(2) Polish Academy of Sciences, Warsaw, Poland

We show that uniformly finite homology of products of $n$ trees vanishes in all degrees except degree $n$, where it is infinite dimensional. Our method is geometric and applies to several large scale homology theories, including almost equivariant homology and controlled coarse homology. As an application we determine group homology with $\ell_{\infty}$-coefficients of lattices in products of trees. We also show a characterization of amenability in terms of 1-homology and construct aperiodic tilings using higher homology.

Keywords: Uniformly finite homology, coarse homology, cohomology of groups, products of trees

Diana Francesca, Nowak Piotr: Eilenberg swindles and higher large scale homology of products of trees. Groups Geom. Dyn. 11 (2017), 371-392. doi: 10.4171/GGD/400