We investigate a set of groups {SBC_n}. Each group SBC_n sits very naturally in the full group of automorphisms of {0,1, ... , n-1}^Z, the full shift on n letters, and is somehow a very natural object. Still, the structure of each group SBC_n, at least initially, was quite a mystery.

These groups' elements are describable as finite transducers, and so the groups SBC_n are linked strongly to the rational group R introduced by Grigorchuk, Nekrashevych, and Suschanski. Furthermore, the group SBC_n corresponds precisely to the outer automorphism group of the generalized Higman-Thompson group G_{n,1} = V_n.

In this talk, we exploit a connection with De Bruijn graphs to begin to explore the structure of the groups SBC_n. Amongst other results, we will show that for m \neq n, SBC_m is not isomorphic to SBC_n, and we will also show that for n>2, the groups SBC_n are infinite, locally finite groups (and hence are not finitely generated).

This talk features work from two separate projects involving collaborators Y. Maissel, A. Navas, and P. Cameron.