Automorphism groups of dense subgroups of R^n

TL;DR

This paper investigates the automorphism groups of dense subgroups of R^n, characterizing them via matrix groups and exploring which subgroups can be realized as automorphism groups, revealing deep connections to number theory.

Contribution

It provides a characterization of automorphism groups of dense subgroups of R^n and explores the realization problem of symmetric subgroups as automorphism groups.

Findings

Aut(G) is identified with I(G) in GL(n,R).

Descriptions of I(G) for many dense subgroups G of R or R^2.

Certain subgroups like {A in GL(n,R): det A=±1} cannot be realized as automorphism groups.

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.