Real Computational Universality: The Word Problem for a Class of Groups with Infinite Presentation

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. As main difference with discrete groups, these groups may be generated by uncountably many generators with index running over certain sets of real numbers. This includes a variety of groups which are not captured by the finite framework of the classical word problem. Our contribution extends computational group theory from the discrete to the BlumShub-Smale (BSS) model of real number computation. It provides a 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 semidecidable (and thus reducible from) but also reducible to the Halting Problem for such machines. It thus gives the first non-trivial example of a problem complete, that is, computationally universal for this model.