Seeking group boundaries for the Farrell-Jones conjecture
by
Frank Quinn
Virginia Tech

The Farrell-Jones conjecture describes algebraic K-theory, L-theory, etc of a group ring in terms of generalized homology with coefficients in K etc of virtually cyclic subgroups. This would provide a very powerful tool for the study of manifolds, stratified sets and some other geometric objects.

It has long seemed likely that this will involve finding a nice boundary for a metric on the group or a related object. Work on other parts of the program has led to a specific proposal. I'll describe this metric, key questions, and some attempts to address them.

Rough version: define a function on G ×G by

p((a, b), (x, y)) = (d(a, x) + d(b, y))/(d(a, b)+d(x, y)+1).

where d is a word metric. Then get a metric from p by integrating along paths and taking the minimum. Roughly this is the standard metric on the diagonal and shrinks as you go away from it. What is the asymptotic structure far from the diagonal?

Date received: January 29, 2008