Proceedings of the London Mathematical Society Advance Access originally published online on October 28, 2008
Proceedings of the London Mathematical Society 2009 98(3):631-651; doi:10.1112/plms/pdn039
| ||||||||||||||||||||||||||||||||||||||||||||||||||
© 2008 London Mathematical Society
Structure and finiteness properties of subdirect products of groups
Mathematics Department
Huxley Building
Imperial College London
London SW7 2AZ
United Kingdom
Current address:
Mathematical Institute
24-29 St Giles'
Oxford OX1 3LB
United Kingdom
Department of Mathematics and Statistics
University of Melbourne
Melbourne 3010
Australia
c.miller@ms.unimelb.edu.au
Received 13 April 2007. Revision received 3 June 2008.
We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNN extensions. We prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of the direct summands. This leads to a characterization of the finitely presented subgroups of the direct product of three free or surface groups and to a solution of the conjugacy problem for arbitrary finitely presented subgroups of direct products of surface groups. We obtain a formula for the first homology of a subdirect product of two free groups and use it to show that there is no algorithm to determine the first homology of a finitely generated subgroup.
2000 Mathematics Subject Classification 20F05, 20E07, 20F67, 20J05, 20F10.
This research was supported by the Australian Research Council, the EPSRC of the United Kingdom, the Swiss National Science Foundation and l’Alliance Scientific (grant #PN 05.004). Author Martin R. Bridson was also supported in part by a Royal Society Nuffield Research Merit Award.