We prove the theorem in the title by constructing an action of a Coxeter group on a product of trees.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm174-1-4, author = {Tadeusz Januszkiewicz}, title = {For Coxeter groups $z^{|g|}$ is a coefficient of a uniformly bounded representation}, journal = {Fundamenta Mathematicae}, volume = {173}, year = {2002}, pages = {79-86}, zbl = {1038.20025}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm174-1-4} }
Tadeusz Januszkiewicz. For Coxeter groups $z^{|g|}$ is a coefficient of a uniformly bounded representation. Fundamenta Mathematicae, Tome 173 (2002) pp. 79-86. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm174-1-4/