Update (july 2009): A detailed posting on the Thompson group appeared on “Geometry and the Imagination,” Danny Calegary’s blog. In spite of two recent preprints one claiming that the Thompson group is amenable and the other claiming the opposite, the problem appears to be open.
We had yesteday, just a day after Independence Day, the annual meeting of the Israeli Mathematical Union and Mati Rubin talked about structures with the property that the automorphism group determines the structure up to isomorphism (even conjugacy). A lovely topic between logic and algebra with relations to many other things. Mati mentioned the Thompson group:
The set of orientation-preserving piecewise-linear homeomorphisms of the unit interval, where the slopes are powers of two and the places where the slope changes are dyadic rationals.
There are many other presentation of the Thompson group. The wikipedia article looks very nice, and there was a special AIM workshop on it in 2004. There are many problems about the Thompson group, and one famous one is: Is it amenable?
Update (May 5): today Yuval Roichman mentioned that Patrick Dehornoy has used the Thompson group in his work regarding the diameter of the graph of the associahedron. This also brings me to the fascinating issue of coincidences that we should discuss sometime.