 
Summary: NEGATIVE CURVATURE AND EFFICIENT
COMPUTATION
JON MCCAMMOND
Abstract. Whenever I am asked to describe geometric group the
ory to a nonspecialist, I tend to explain a theorem due to Gromov
which relates the groups with an intrinsic geometry like that of
the hyperbolic plane to those in which certain computations can
be efficiently carried out. This theorem was one of the clearest
early indications that applying a metric perspective to traditional
group theory problems might lead to new and important insights.
The theorem that I want to talk about asserts that there is a close
relationship between two collections of groups: one collection is de
fined geometrically and the other is defined computationally. The first
section describes the relevant geometric and topological ideas, the sec
ond discusses the key algebraic and computational concepts, and the
short final section describes the relationship between them. An oral
presentation style is maintained throughout.
1. Geometry and topology
The first thing to highlight is that there is close relationship between
groups and topological spaces. More specifically, to each connected
