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- W4229865670 abstract "This chapter discusses the notion of space, first by explaining what it means for a group to be a group of symmetries of a geometric object. This is the idea of group action, and some examples are given. The chapter proceeds by defining, for any group <italic>G</italic>, the Cayley graph of <italic>G</italic> and shows that the symmetric group of of this graph is precisely the group <italic>G</italic>. It then introduces metric spaces, which formalize the notion of a geometric object, and highlights numerous metric spaces that groups can act on. It also demonstrates that groups themselves are metric spaces; in other words, groups themselves can be thought of as geometric objects. The chapter concludes by using these ideas to frame the motivating questions of geometric group theory. Exercises relevant to each idea are included." @default.
- W4229865670 created "2022-05-11" @default.
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- W4229865670 date "2017-07-11" @default.
- W4229865670 modified "2023-09-25" @default.
- W4229865670 title "and Spaces" @default.
- W4229865670 doi "https://doi.org/10.23943/princeton/9780691158662.003.0002" @default.
- W4229865670 hasPublicationYear "2017" @default.
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