Quasi-actions and rough Cayley graphs for locally compact groups

Details Quasi-actions and rough Cayley graphs for locally compact groups Quasi-actions and rough Cayley graphs for locally compact groups

2011-12-29

arxiv.org/abs/1112.6415v1

Mathematics

Group Theory

13 pages

Scientific paper

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 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 growth (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.

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