Totally disconnected groups from Baumslag-Solitar groups

TL;DR

This paper constructs totally disconnected locally compact groups from Baumslag-Solitar groups, analyzing their scale function, flat rank, and subgroup structure to reveal new topological and algebraic properties.

Contribution

It introduces a method to embed Baumslag-Solitar groups into totally disconnected groups and studies their invariants and subgroup structures.

Findings

Scale function distinguishes parameters m and n

G_{m,n} has flat rank 1 or 0 depending on |m| and |n|

G_{m,n} contains a compact open subgroup isomorphic to a product of p-adic integers

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|)^{\rho}, (lcm(m,n)/|n|)^{\rho} | \rho\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}.