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Any finitely generated group can be turned into a proper metric space using awordmetric.
From theCambridge English Corpus
Given two elements, one constructs thewordmetricgiven by the length of the minimal path between the elements.
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Wikipedia
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This action is an isometry of thewordmetric.
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Formally, this means classifying finitely generated groups with theirwordmetricup to quasi-isometry.
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The distance between 0 and -3 in thewordmetricis therefore equal to 3.
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This example is from Wikipedia and may be reused under a CC BY-SA license.
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