Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups

Details Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups

2011-11-26

arxiv.org/abs/1111.6181v1

Mathematics

Group Theory

Scientific paper

Given an automorphism $\phi:\Gamma\lr \Gamma$, one has an action of $\Gamma$ on itself by $\phi$-twisted conjugacy, namely, $g.x=gx\phi(g^{-1})$. The orbits of this action are called $\phi$-twisted conjugacy classes. One says that $\Gamma$ has the $R_\infty$-property if there are infinitely many $\phi$-twisted conjugacy classes for every automorphism $\phi$ of $\Gamma$. In this paper we show that $\SL(n,\bz)$ and its congruence subgroups have the $R_\infty$-property. Further we show that any (countable) abelian extension of $\Gamma$ has the $R_\infty$-property where $\Gamma$ is a torsion free non-elementary hyperbolic group, or $\SL(n,\bz), \Sp(2n,\bz)$ or a principal congruence subgroup of $\SL(n,\bz)$ or the fundamental group of a complete Riemannian manifold of constant negative curvature.

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