Solving the word problem in real time

The paper is devoted to the study of groups whose word problem can be solve d by a Turing machine which operates in real time. A recent result of the f irst author for word hyperbolic groups is extended to prove that under cert ain conditions the generalised Dehn algorithms of Cannon, Goodman and Shapi ro, which clearly run in linear time, can be programmed on real-time Turing machines. It follows that word-hyperbolic groups, finitely generated nilpo tent groups and geometrically finite hyperbolic groups all have real-time w ord problems.