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*by a standard (one-tape) Turing machine. It is well-known that the wordproblem for hyperbolic groups, whence in particular for free groups, can besolved in linear time. However, these algorithms run on machines morecomplicated than a standard Turing machine. By contrast, in this note we showthat a standard Turing machine cannot solve the word problem for the free groupon two generators in less than quadratic time.

The word problem for free groups cannot be solved in linear time*