Eilenberg swindles and higher large scale homology of products of trees

TL;DR

This paper investigates the large scale homology of products of trees, revealing vanishing and infinite dimensionality in specific degrees, and applies these results to group homology and tiling constructions.

Contribution

It introduces a geometric method to analyze large scale homology of product of trees, extending to various homology theories and applications.

Findings

Homology vanishes in all degrees except degree n for products of n trees.

Group homology with ℓ∞-coefficients of lattices in products of trees is characterized.

Constructs aperiodic tilings using higher homology.

Abstract

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.