Commentarii Mathematici Helvetici

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Volume 84, Issue 2, 2009, pp. 259–296
DOI: 10.4171/CMH/162

Published online: 2009-06-30

Groups with finitely many conjugacy classes and their automorphisms

Ashot Minasyan[1]

(1) University of Southampton, United Kingdom

We combine classical methods of combinatorial group theory with the theory of small cancellation over relatively hyperbolic groups to construct finitely generated torsion-free groups that have only finitely many classes of conjugate elements. Moreover, we present several results concerning embeddings into such groups.

As another application of these techniques, we prove that every countable group C can be realized as a group of outer automorphisms of a group N, where N is a finitely generated group having Kazhdan’s property (T) and containing exactly two conjugacy classes.

Keywords: Conjugacy classes, relatively hyperbolic groups, outer automorphism groups

Minasyan Ashot: Groups with finitely many conjugacy classes and their automorphisms. Comment. Math. Helv. 84 (2009), 259-296. doi: 10.4171/CMH/162