Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups

Mubeena, T. ; Sankaran, P.

Canadian mathematical bulletin, Tome 57 (2014) no. 1, pp. 132-140

Résumé

Given a group automorphism $\phi :\,\Gamma \,\to \,\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 $\text{SL(}n\text{,}\mathbb{Z}\text{)}$ 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 $\text{SL(}n\text{,}\mathbb{Z}\text{)},\text{Sp(2}n\text{,}\mathbb{Z}\text{)}$ or a principal congruence subgroup of $\text{SL(}n\text{,}\mathbb{Z}\text{)}$ or the fundamental group of a complete Riemannian manifold of constant negative curvature.

Détail

DOI : 10.4153/CMB-2012-013-7

Mots-clés : 20E45, twisted conjugacy classes, hyperbolic groups, lattices in Lie groups

Bibliographie

[1] [1] Brauer, R., Representations of finite groups. In: 1963 Lectures on Modern Mathematics, Vol. I, Wiley, New York, 1963, pp. 133–175. Google Scholar

[2] [2] Bridson, M. R. and Haefliger, A. Metric spaces of non-positive curvature. Grundlehren der MathematischenWissenschaften, 319, Springer-Verlag, Berlin, 1999. Google Scholar

[3] [3] Fel'shtyn, A. L., The Reidemeister number of any automorphism of a Gromov hyperbolic group isinfinite. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 279 (2001), Geom. i Topol., 6, 229–240, 250; translation in Math. Sci. (N.Y.) 119 (2004), no. 1, 117–123. Google Scholar

[4] [4] Fel'shtyn, A. L., New directions in Nielsen–Reidemeister theory. Topology Appl. 157 (2010), no. 10–11, 1724–1735. Google Scholar | DOI

[5] [5] Fel’shtyn, A. and Gonçalves, D. L., Twisted conjugacy classes in symplectic groups, mapping classgroups and braid groups. With an appendix written jointly with Francois Dahmani. Geom. Dedicata 146 (2010), 211–223. Google Scholar | DOI

[6] [6] Fel’shtyn, A. and Hill, R., The Reidemeister zeta function with applications to Nielsen theory and aconnection with Reidemeister torsion. K-Theory 8 (1994), no. 4, 367–393. Google Scholar | DOI

[7] [7] Gonçalves, D. L. and Wong, P., Twisted conjugacy classes in nilpotent groups. J. Reine Angew. Math. 633 (2009), 11–27. Google Scholar | DOI

[8] [8] Hua, L. K. and Reiner, I., Automorphisms of the unimodular group. Trans. Amer. Math. Soc. 71 (1951), 331–348. Google Scholar | DOI

[9] [9] Levitt, G. and Lustig, M., Most automorphisms of a hyperbolic group have very simple dynamics. Ann. Sci. Ecol. Norm. Sup. 33 (2000) 507–517. Google Scholar

[10] [10] Lyndon, R. and Schupp, P., Combinatorial group theory. Reprint of the 1977 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Google Scholar

[11] [11] Nasybullov, T. R., Twisted conjugacy classes in special and general linear groups. arxiv:1201.6515. Google Scholar

[12] [12] Newman, M., Normalizers of modular groups. Math. Ann. 238 (1978), no. 2, 123–129. Google Scholar | DOI

[13] [13] O'Meara, O. T., The automorphisms of the linear groups over any integral domain. J. Rein. Angew. Math. 223 (1966), 56–100. Google Scholar

[14] [14] Osin, D., Small cancellations over relatively hyperbolic groups and embedding theorems. Ann. of Math. (2) 172 (2010), no. 1, 1–39. Google Scholar | DOI

[15] [15] Pyber, L., Finite groups have many conjugacy classes. J. London Math. Soc. 46 (1992), no. 2, 239–249. Google Scholar | DOI

[16] [16] Raghunathan, M. S., Discrete subgroups of Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, 68, Springer-Verlag, New York-Heidelberg, 1972. Google Scholar

[17] [17] Reiner, I., Automorphisms of the symplectic modular group. Trans. Amer. Math. Soc. 80 (1955), 35–50. Google Scholar | DOI

[18] [18] Sela, Z., Endomorphisms of hyperbolic groups. I. The Hopf property. Topology 38 (1999), no. 2, 301–321. Google Scholar | DOI

[19] [19] Serre, J.-P., Trees. Translated from the French original by John Stillwell. Corrected 2nd printing of the 1980 English translation, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Google Scholar

[20] [20] Sury, B., Congruence subgroup property. An elementary approach aimed at applications. Texts and Readings in Mathematics, 24, Hindustan Book Agency, New Delhi, 2003. Google Scholar

[21] [21] Zimmer, R. J., Ergodic theory and semisimple Lie groups. Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984. Google Scholar