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

^{1}

^{2}

^{*}

## Abstract

**:**

**Theorem**

**1.**

- (i)
- if $0<k<\omega $, then $lg\left(x\right)\le lg\left({x}^{k}\right)$;
- (ii)
- if $lg\left(y\right)<k<\omega $ and ${x}^{k}=y$, then $x=e$.

**Proof**

**of**

**Theorem**

**1.**

- (1)
- ${f}_{1}\left(n\right)>lg\left({g}_{n}^{\prime}\right)$;
- (2)
- ${f}_{2}\left(n\right)=({m}_{*}+1)+{\sum}_{\ell <n}{f}_{1}\left(\ell \right)$.

**Claim**

**1.**

**Proof.**

**Definition**

**1.**

**Definition**

**2.**

**Fact**

**1.**

- (1)
- g can be written as $hf{h}^{-1}$ with f cyclically reduced and $lg\left(g\right)=lg\left(f\right)+2lg\left(h\right)$;
- (2)
- if $0<k<\omega $ and f is cyclically reduced, then $lg\left({f}^{k}\right)=klg\left(f\right)$;
- (3)
- if $0<k<\omega $ and $g=hf{h}^{-1}$ is as in (1), then $lg{\left(hf{h}^{-1}\right)}^{k}=klg\left(f\right)+2lg\left(h\right)$.

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

- (i)
- ${\left(ab\right)}^{2}=1$;
- (ii)
- $lg\left(ab\right)=2<3$, ${\left(ab\right)}^{3}=ab$ and $ab\ne e$.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Shelah, S. A Countable Structure Does Not Have a Free Uncountable Automorphism Group. Bull. Lond. Math. Soc.
**2003**, 35, 1–7. [Google Scholar] [CrossRef] - Becker, H.; Kechris, A.S. The Descriptive Set Theory of Polish Group Actions; London Math. Soc. Lecture Notes Ser. 232; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar]
- Shelah, S. Polish Algebras, Shy From Freedom. Israel J. Math.
**2011**, 181, 477–507. [Google Scholar] [CrossRef] - Solecki, S. Polish Group Topologies. In Sets and Proofs; London Math. Soc. Lecture Note Ser. 258; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar]
- Dudley, R.M. Continuity of Homomorphisms. Duke Math. J.
**1961**, 28, 34–60. [Google Scholar] [CrossRef] - Paolini, G.; Shelah, S. Polish Topologies for Graph Products of Cyclic Groups. In Preparation.
- Servatius, H. Automorphisms of Graph Groups. J. Algebra
**1989**, 126, 34–60. [Google Scholar] [CrossRef]

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Paolini, G.; Shelah, S.
No Uncountable Polish Group Can be a Right-Angled Artin Group. *Axioms* **2017**, *6*, 13.
https://doi.org/10.3390/axioms6020013

**AMA Style**

Paolini G, Shelah S.
No Uncountable Polish Group Can be a Right-Angled Artin Group. *Axioms*. 2017; 6(2):13.
https://doi.org/10.3390/axioms6020013

**Chicago/Turabian Style**

Paolini, Gianluca, and Saharon Shelah.
2017. "No Uncountable Polish Group Can be a Right-Angled Artin Group" *Axioms* 6, no. 2: 13.
https://doi.org/10.3390/axioms6020013