NAME:      Ross Geoghegan  (State University of New York at Binghamton)

TITLE:     SL(2) actions on the hyperbolic plane

ABSTRACT:

Take a positive integer m and consider the action of SL(2, Z[1/m]) on the
hyperbolic plane by Moebius transformations.  More generally, consider a
not-necessarily-discrete action of a discrete group G on a non-positively
curved (i.e. CAT(0)) space M.  Are there interesting topological invariants
of such an action - invariants which distinguish one from another?

I will describe a new kind of "controlled topology" invariant of
such an action - a measure of how highly connected the action is.
In particular, for the SL(2, Z[1/m]) case it will turn out that
the action is (s-2)-connected but not (s-1)-connected, where s is
the number of different primes which divide m.  The most interesting
feature of the proof of this is the role played by the Borel subgroup
of upper triangular matrices.  (Joint work with Robert Bieri.)