Automorphism groups of dense subgroups of R^n

Abstract

By an automorphism of a topological group G we mean an isomorphism of G onto itself which is also a homeomorphism. In this article, we study the automorphism group Aut(G) of a dense subgroup G of R^n, n>=1. We show that Aut(G) can be naturally identified with the subgroup I(G)={A in GL(n,R): G A =G} of the group GL(n,R) of all non-degenerated (n x n)-matrices over R, where G A={g A:g in G}. We describe $I(G) for many dense subgroups G of either R or R^2. We consider also an inverse problem of which symmetric subgroups of GL(n,R) can be realized as I(G) for some dense subgroup G of R^n. For example, for n>=2, we show that the group {A in GL(n,R): det A=+-1} cannot be realized in this way. The realization problem is quite non-trivial even in the one-dimensional case and has deep connections to number theory.