# No Uncountable Polish Group Can be a Right-Angled Artin Group

## TL;DR

This paper demonstrates that uncountable Polish groups cannot have a generator system with a length function satisfying specific properties, implying such groups cannot be automorphism groups of uncountable structures, extending previous results.

## Contribution

It proves that uncountable Polish groups cannot be right-angled Artin groups, generalizing earlier results for free and free Abelian groups.

## Key findings

- Uncountable Polish groups cannot admit certain length functions.
- Automorphism groups of countable structures are not uncountable right-angled Artin groups.
- Extends previous results from free and free Abelian groups.

## Abstract

We prove that no uncountable Polish group can admit a system of generators whose associated length function satisfies the following conditions: (i) if $0 < k < \omega$, then $lg(x) \leq lg(x^k)$; (ii) if $lg(y) < k < \omega$ and $x^k = y$, then $x = e$. In particular, the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes results from [3] and [5], where this is proved for free and free Abelian uncountable groups.