The Automorphism Group of a Free-by-Cyclic Group in Rank 2

Abstract

Let φ be an automorphism of a free group F_n of rank n, and let M_φ = F_n ⋊_φ ℤ be the corresponding mapping torus of φ. We study the group Out(M_φ) under certain technical conditions on φ. Moreover, in the case of rank 2, we classify the cases when this group is finite or virtually cyclic, depending on the conjugacy class of the image of φ in GL₂(ℤ). As an application, we solve the isomorphism problem for the family of F₂-by-ℤ groups, in terms of the two defining automorphisms.

Keywords:

• Free-by-cyclic group
• Free group
• Isomorphism problem

Mathematics Subject Classification:

• Primary 20F28
• Secondary 20E22

ACKNOWLEDGMENTS

The first author is partially supported by the grant of the President of Russian Federation for young Doctors MD-326.2003.01, by the INTAS grant N 03-51-3663, and by the Centre de Recerca Matemàtica (CRM) at Barcelona. The second author gratefully acknowledges the support of the CRM and the Universitat Politèctnica de Catalunya (UPC). The third author is partially supported by DGI (Spain) through grant BFM2003-06613, and by the Generalitat de Catalunya through grant ACI-013. The three authors thank the CRM for his hospitality during the academic course 2004–2005, while this paper has been written.

Notes

Communicated by A. Facchini.