ALPS’07 – Groups and Complexity

The connection between groups and recursive (un)decidability has a long history, going back to the early 1900s. Also, various polynomial-time algorithms have been known in group theory for a long time. However the impact of more general computational complexity (e.g., NP-completeness or PSpace-completeness) has been relatively small and recent. These lectures review a sampling of older facts about algorithmic problems in group theory, and then present more recent results about the connection with complexity: isoperimetric functions and NP; Thompson groups, boolean circuits, and coNP; Thompson monoids and circuit complexity; Thompson groups, reversible computing, and #P; distortion of Thompson groups within Thompson monoids, and one-way permutations. We are especially interested in deep connections between computational complexity and group theory. By “connection” we do not just mean analyzing the computational complexity of algorithms about groups. We are more interested in algebraic characterizations of complexity classes in terms of group theory, i.e., in finding a “mirror image” of all of complexity theory within group theory. Conversely, we are interested in the computational nature of concepts that appear at first purely algebraic.