# Characterizing Lie groups by controlling their zero-dimensional subgroups

## TL;DR

This paper characterizes Lie groups by their zero-dimensional subgroups, identifying conditions under which various topological groups are Lie groups based on properties like compactness, minimality, and zero-dimensional subgroup structure.

## Contribution

It provides new characterizations of Lie groups using the properties of their zero-dimensional metric subgroups, extending understanding of their topological structure.

## Key findings

- A Lie group is locally compact with no infinite compact metric zero-dimensional subgroups.
- An abelian Lie group is characterized by local minimality, precompactness, and discrete zero-dimensional subgroups.
- A compact Lie group is minimal with no infinite closed metric zero-dimensional subgroups.

## Abstract

We provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The "compact-like" properties we consider include (local) compactness, (local) omega-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is a sample of our characterizations: (i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups. (ii) An abelian topological group G is a Lie group if and only if G is locally minimal, locally precompact and all closed metric zero-dimensional subgroups of G are discrete. (iii) An abelian topological group is a compact Lie group if and only if it is minimal and has no infinite closed metric zero-dimensional subgroups. (iv) An infinite topological group is a compact Lie group if and only if it is sequentially complete, precompact, locally minimal, contains a non-empty open connected subset and all its compact metric zero-dimensional subgroups are finite.