Sol geometry groups are not asynchronously automatic (Q2766411)

Automatic groups can be described by finite state automata and have an efficient algorithm for solving the word problem. An automatic structure imposes certain restrictions on the geometry of the group, for example a quadratic isoperimetric inequality, and the concept of an asynchronously automatic group is an attempt to broaden the class of automatic groups. The fundamental group of a closed 3-manifold which satisfies Thurston's geometrization conjecture is automatic if and only if none of the prime components of the manifold are modeled on the geometries Nil or Sol; also, Sol manifold groups are not automatic, and closed Nil manifolds are not even asynchronously automatic. It is the main result of the present paper that lattices in the 3-dimensional Lie group Sol are not asynchronously automatic (or, in a stronger version, do not admit a regular, asynchronous combing with uniqueness). ``The proof of the Theorem will have to use something more than just the geometry of an asynchronous combing. The extra ingredient used is the fact that the combing associated to an asynchronously automatic structure has to be a regular language''.