Some properties of the conjugacy class growth function. (English) Zbl 1071.20504

Rivin, Igor

Myasnikov, A.G. (ed.) et al., Group theory, statistics, and cryptography. AMS special session combinatorial and statistical group theory, New York University, NY, USA, April 12–13, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3444-4/pbk). Contemporary Mathematics 360, 113-117 (2004).

Summary: We study the number of conjugacy classes of bounded length in infinite groups, concentrating on free groups. We show that the derivative of the growth function has an infinite number of poles in the unit disk, in particular, it (the derivative) is not rational and neither, of course, is the function itself.
For the entire collection see [Zbl 1053.20500].

MSC:

20F05	Generators, relations, and presentations of groups
20E45	Conjugacy classes for groups
20E05	Free nonabelian groups
20F67	Hyperbolic groups and nonpositively curved groups

Keywords:

finitely presented groups; numbers of conjugacy classes; word-hyperbolic groups; free groups; growth functions