Negative curvature and residual finiteness
Abstract: Gromov revolutionized infinite group theory by treating groups as geometric objects. From this viewpoint, the most prominent class of groups are the word-hyperbolic groups, which are characterized by being coarsely negatively curved. In the last 10 years, word-hyperbolic groups have recieved a tremendous amount of attention, and many of their most important properties have been understood.
One striking problem which remains open is whether or not every word-hyperbolic group is residually finite. A group G is called residually finite provided that for each non-trivial element g of G, there is a finite quotient p:G -> Q such that p(g) is not trivial. Some well known groups which are residually finite include surface groups, and polycyclic groups and more generally linear groups. Also, the fundamental group of any compact 3-manifold satisfying Thurston's geometrization conjecture is residually finite. It is widely believed that there do exist word-hyperbolic groups which are not residually finite. However if these groups do exist, they have been hiding from everybody for a long time!
In my talk, I will survey some evidence for and against the residual finiteness of word-hyperbolic groups. I will describe examples of non-residually finite groups which are non-positively curved. And I will discuss some of my more recent work which suggests that there is indeed a connection between negative curvature and residual finiteness.