#### Volume 18, issue 6 (2018)

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Action dimension of lattices in Euclidean buildings

### Kevin Schreve

Algebraic & Geometric Topology 18 (2018) 3257–3277
##### Abstract

We show that if a discrete group $\Gamma$ acts properly and cocompactly on an $n$–dimensional, thick, Euclidean building, then $\Gamma$ cannot act properly on a contractible $\left(2n-1\right)$–manifold. As an application, if $\Gamma$ is a torsion-free $S$–arithmetic group over a number field, we compute the minimal dimension of contractible manifold that admits a proper $\Gamma$–action. This partially answers a question of Bestvina, Kapovich, and Kleiner.

##### Keywords
action dimension, Euclidean building, S-arithmetic group, van Kampen obstruction
##### Mathematical Subject Classification 2010
Primary: 20F36, 20F65, 20F55, 57Q35
Secondary: 20J06
##### Publication
Received: 7 May 2017
Revised: 4 March 2018
Accepted: 17 April 2018
Published: 18 October 2018
##### Authors
 Kevin Schreve Department of Mathematics University of Michigan Ann Arbor, MI United States https://sites.google.com/site/kevinschreve/