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TWISTED CONJUGACY CLASSES IN EXPONENTIAL GROWTH GROUPS

Published online by Cambridge University Press:  20 March 2003

DACIBERG GONÇALVES  and
PETER WONG
DACIBERG GONÇALVES
Affiliation:
Dept. de Matemática – IME – USP, Caixa Postal 66.281, CEP 05315-970, São Paulo – SP, Brazildlgoncal@ime.usp.br
PETER WONG
Affiliation:
Department of Mathematics, Bates College, Lewiston, ME 04240, USApwong@abacus.bates.edu
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Abstract

Let $\phi\,{:}\,G \to G$ be a group endomorphism where G is a finitely generated group of exponential growth, and denote by $R(\phi)$ 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 $R({\phi}^n)\,{<}\,\infty$ for all n.

Keywords

20E4537C2520F1655M2020F99
Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 35 , Issue 2 , March 2003 , pp. 261 - 268
Copyright
© The London Mathematical Society 2003

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Footnotes

This work was completed during the second author's visit to São Paulo October 12–19 2001. The visit was partially supported by a grant from Bates College and the ‘Projeto temático Topologia Algébrica e Geométrica-FAPESP’.