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Twisted Conjugacy Classes in Exponential Growth Groups
Author(s) -
Gonçalves Daciberg,
Wong Peter
Publication year - 2003
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609302001832
Subject(s) - mathematics , conjugacy class , exponential function , exponential growth , pure mathematics , mathematics education , mathematical analysis
Let φ: G → G be a group endomorphism where G is a finitely generated group of exponential growth, and denote by R (φ) the number of twisted φ‐conjugacy classes. Fel'shtyn and Hill ( K‐theory 8 (1994) 367–393) conjectured that if φ is injective, then R(φ) is infinite. This paper shows that this conjecture does not hold in general. In fact, R(φ) can be finite for some automorphism φ. Furthermore, for a certain family of polycyclic groups, there is no injective endomorphism φ with R (φ n ) < ∞ for all n . 2000 Mathematics Subject Classification 20E45, 37C25, 20F16, 55M20, 20F99.