Basic Notions Seminar talk on October 4, 2002

Speaker:   Theodore Palmer (University of Oregon)

Title:   Totally Disconnected Groups

Abstract:  The connected component $G_e$ of any topological group $G$ is a closed normal subgroup and gives a short exact sequence

        The last two thirds of the Twentieth Century saw intensive study of connected groups using connected Lie groups to which they are closely related. They are reasonably well understood.

        Every group (finite or infinite) is a topological group under the discrete topology and hence a totally disconnected group. Thus we cannot expect to say anything interesting about totally disconnected groups without some further restrictions.

        The strongest restriction would be to consider compact, totally disconnected groups. It turns out that these are exactly the topological groups which are projective limits of finite groups: the pro-finite groups. I will discuss these using the example of the Galois group of an infinite extension of a field, e.g. the algebraic numbers over the rational numbers.

        Compact totally disconnected groups were reasonably well understood by the middle of the Twentieth Century. However, a number of real experts were on record as saying that there was no hope of understanding much in general about locally compact totally disconnected groups until George A. Willis proved them completely wrong in 1994. I will outline Willis' theory.