Geometric group theory in need of algebraic topology

In this expository talk I will
1. introduce Thompson’s Group \$F\$, a famous finitely generated group.
2. explain the general concept of quasi-isometry of finitely generated groups;
3. state an unsolved problem, namely: is F quasi-isometric to \$F \times Z\$?
4. explain why this is an infinite-dimensional or “phantom map” type of problem so that ordinary invariants will not detect the difference between the two groups (if there is a difference).
If time permits, I will discuss two older problems in geometric topology which exhibit some similarities to this one, and which needed reduced complex K-theory or the Sullivan Conjecture for their solution.