ABSTRACT
In this chapter, we consider a number of topics that are special to the hyperbolic groups.
The symbols ∗632, ∗623, ∗362, ∗326, and ∗263, ∗236 all mean the same thing. What’s the general rule for telling when two signatures are “really the same”? For the reasons explained in Chapter 3, “the same” should mean that the corresponding groups are isotopic. The answer is best expressed by describing various operations that don’t change the isotopy type. Namely, the group represented by the typical signature
. . .◦◦◦ABC ... ∗a1b1c1...∗a2b2c2... ∗ anbncn...×× will be unchanged up to isotopy if we
• exchange an ◦ and an × for three ×’s, • freely permute the digits A,B,C,... that correspond to gyrations,
• cyclically permute the digits ak,bk, ck, ... in any one kaleidoscope,
• freely permute the portions ∗a1b1c1 . . . , ∗a2b2c2 . . . , . . . , ∗anbncn... of the signature corresponding to the individual kaleidoscopes,
• simultaneously reverse the cyclic orders in all n kaleidoscopic portions.
