Metric properties of the lamplighter group as an automata group

We examine the geometry of the Cayley graph of the lamplighter group with respect to the generating set rising from its interpretation as an automata group due to Grigorchuk and Zuk. We find some metric behavior with respect to this generating set analogous to the metric behavior in the standard wreath product generating set. The similar metric behavior includes expressions for geodesic paths and families of dead-end' elements, which are endpoints of terminating geodesic rays. We also exhibit some different metric behavior between these two generating sets related to the existence of seesaw’ elements.