256 research outputs found
Geometry of generated groups with metrics induced by their Cayley color graphs
Let be a group and let be a generating set of . In this article,
we introduce a metric on with respect to , called the cardinal
metric. We then compare geometric structures of and ,
where denotes the word metric. In particular, we prove that if is
finite, then and are not quasi-isometric in the case when
has infinite diameter and they are bi-Lipschitz equivalent
otherwise. We also give an alternative description of cardinal metrics by using
Cayley color graphs. It turns out that color-permuting and color-preserving
automorphisms of Cayley digraphs are isometries with respect to cardinal
metrics
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