The word problem for discrete groups is well-known to be undecidable by a
Turing Machine; more precisely, it is reducible both to and from and thus
equivalent to the discrete Halting Problem.
The present work introduces and studies a real extension of the word problem
for a certain class of groups which are presented as quotient groups of a free
group and a normal subgroup. Most important, the free group will be generated
by an uncountable set of generators with index running over certain sets of
real numbers. This allows to include many mathematically important groups which
are not captured in the framework of the classical word problem.
Our contribution extends computational group theory from the discrete to the
Blum-Shub-Smale (BSS) model of real number computation. We believe this to be
an interesting step towards applying BSS theory, in addition to semi-algebraic
geometry, also to further areas of mathematics.
The main result establishes the word problem for such groups to be not only
semi-decidable (and thus reducible FROM) but also reducible TO the Halting
Problem for such machines. It thus provides the first non-trivial example of a
problem COMPLETE, that is, computationally universal for this model.Comment: corrected Section 4.