Totally disconnected groups from Baumslag-Solitar groups

Abstract

For each Baumslag-Solitar group BS(m,n) (m,n nonzero integers), a totally disconnected, locally compact group, G_{m,n}, is constructed so that BS(m,n) is identified with a dense subgroup of G_{m,n}. The scale function on G_{m,n}, a structural invariant for the topological group, is seen to distinguish the parameters m and n to the extent that the set of scale values is {(lcm(m,n)/|m|)^ρ, (lcm(m,n)/|n|)^ρ | ρ\in N}. It is also shown that G_{m,n} has flat rank 1 when |m|\neq |n| and 0 otherwise, and that G_{m,n} has a compact, open subgroup isomorphic to the product {(Z_p,+) | p is a prime divisor of the scale}.