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

Gianluca Paolini,Saharon Shelah

doi:10.3390/axioms6020013

Gianluca Paolini, Saharon Shelah

Open Access

https://doi.org/10.3390/axioms6020013

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Abstract

We prove that if G is a Polish group and A a group admitting a system of generators whose associated length function satisfies: (i) if 0 < k < ω , then l g ( x ) ≤ l g ( x k ) ; (ii) if l g ( y ) < k < ω and x k = y , then x = e , then there exists a subgroup G * of G of size b (the bounding number) such that G * is not embeddable in A. In particular, we prove that the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes analogous results for free and free abelian uncountable groups.

Highlights

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J
Inspired by the question of Becher and Kechris, Solecki [4] proved that no uncountable Polish group can be free abelian
We give a general framework for these results, proving that no uncountable Polish group can be a right-angled Artin group

Summary

Introduction

Inspired by the question of Becher and Kechris, Solecki [4] proved that no uncountable Polish group can be free abelian. We give a general framework for these results, proving that no uncountable Polish group can be a right-angled Artin group (see Definition 1). Let G = ( G, d) be an uncountable Polish group and A a group admitting a system of generators whose associated length function satisfies the following conditions: (i) if 0 < k < ω, lg( x ) ≤ lg( x k ); (ii) if lg(y) < k < ω and x k = y, x = e.

Results

Conclusion