Quasi-actions and rough Cayley graphs for locally compact groups [article]

Pekka Salmi
2013 arXiv   pre-print
We define the notion of rough Cayley graph for compactly generated locally compact groups in terms of quasi-actions. We construct such a graph for any compactly generated locally compact group using quasi-lattices and show uniqueness up to quasi-isometry. A class of examples is given by the Cayley graphs of cocompact lattices in compactly generated groups. As an application, we show that a compactly generated group has polynomial growth if and only if its rough Cayley graph has polynomial
more » ... (same for intermediate and exponential growth). Moreover, a unimodular compactly generated group is amenable if and only if its rough Cayley graph is amenable as a metric space.
arXiv:1112.6415v2 fatcat:adxxjg34zre6ll6mhoenjryweq