Expansive maps of the circle
by Chris Connell, Steven Hurder, and Alex Furman

HISTORY:

It has been known for some time that an action of an abelian group on the circle cannot be expansive. Tom Ward raised the question at Seattle (in July 1999) whether this result extends to nilpotent groups as well.

The three authors began discussing the problem in March 2000, and over the course of a week found a proof that a group which acts expansively must have exponential growth, hence cannot be nilpotent.

We subsequently learned that Ralf Spatzier had also found a proof after the Seattle Conference, which has now been written up. Spatzier's proof is more algebraic, but much shorter (one page!)

However, literature research revealed that T. Inaba and N. Tsuchiya had shown many years ago that an expansive foliation of codimension one must have a resilient leaf, and their proof covers the topological case. (cf. Expansive Foliations, Hokkaido Math. Journal, 21:39--49, 1992.) When applied to the suspension of a topological group action on a circle, this gives a proof that if a group G admits an expansive action on the circle, then the Cayley graph of G has exponential growth, hence is not nilpotent.