Brin's Groups nV
Time and place
1 PM on Thursday, March 26th, 2009; NAC 6/113
Dr. Daniel S. Farley (Miami University of Ohio)
Abstract
In the 1960s, Richard Thompson introduced a group V, which is a certain group of piecewise-linear homeomorphisms
of the Cantor set. Thompson's group V was one of the first examples of a finitely presented infinite simple group (the earliest
example was a related group T, also defined by Thompson). In the 1980s, Brown and Geoghegan showed that T and V have type
F-infinity, i.e., there are aspherical cell complexes K(T,1) and K(V,1) having only finitely many cells in each dimension, and having
T and V (respectively) as their fundamental groups. In 2003, I was able to show that T and V act properly on locally compact, contractible, non-positively curved cubical complexes (that is, on proper CAT(0) cubical complexes).
Matthew Brin defined generalizations of Thompson's group V, which he calls nV. The group nV is a certain group of
piecewise-linear homeomorphisms of the n-fold product of the Cantor set. Brin is able to show that V is not isomorphic to 2V,
and he computes a finite presentation for 2V.
In this talk, I will describe joint work with Bruce Hughes. We build proper CAT(0) cubical complexes on which the groups nV
act properly by isometries. Hughes has a more general theory of groups acting by local similarities on compact ultrametric
spaces, and all of the groups nV are special cases. I will describe some of this theory as time permits. I will also talk about
the problem of determining whether the groups nV have type F-infinity.
