Martin Bridson: Unsolvable decision problems for discrete and profinite groups

Using the Higman Embedding Theorem and old ideas of Adian and Rabin, one can prove in a straightforward way that many decision problems for finitely presented groups are unsolvable. But certain natural problems do not lend themselves to this template, and as a result the question of whether they are algorithmically decidable or not has remained unresolved until recently. Such problems include the question of whether a group has a non-trivial linear representation, or a linear representation with infinite image, or whether the group is large (i.e.has a subgroup of finite that maps onto a free group).

In this talk I shall describe recent work with Henry Wilton in which we prove that all of these problems are undecidable, even among the fundamental groups of compact non-positively curved 2-complexes.