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Journal of the London Mathematical Society 2001 63(3):623-639; doi:10.1017/S0024610701002083
© 2001 by London Mathematical Society
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© The London Mathematical Society

Solving the Word Problem in Real Time

Derek F. Holt and Sarah Rees

Mathematics Institute, University of Warwick Coventry CV4 7AL, dfh{at}maths.warwick.ac.uk
Department of Mathematics, University of Newcastle Newcastle NE1 7RU, Sarah.Rees{at}ncl.ac.uk

Received 11 January 2000.

The paper is devoted to the study of groups whose word problem can be solved by a Turing machine which operates in real time. A recent result of the first author for word hyperbolic groups is extended to prove that under certain conditions the generalised Dehn algorithms of Cannon, Goodman and Shapiro, which clearly run in linear time, can be programmed on real-time Turing machines. It follows that word-hyperbolic groups, finitely generated nilpotent groups and geometrically finite hyperbolic groups all have real-time word problems.


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