A non-polybounded absolutely closed $36$-Shelah group

Abstract

For every infinite cardinal $\kappa$ with ${\kappa }^{+}=2^{\kappa }$ we construct a group $G$ of cardinality $|G|={\kappa }^{+}$ such that (i) $G$ is $36$-Shelah, which means that $A^{36}=G$ for any subset $A\subseteq G$ of cardinality $|A|=|G|$; (ii) $G$ is absolutely ${\mathsf{T}}_1\mathsf{S}$-closed and projectively ${\mathsf{T}}_1\mathsf{S}$-discrete, which means that for every homomorphism $h:G\to Y$ to a $T_1$ topological semigroup $Y$ the image $h[G]$ is a closed discrete subspace of $Y$, (iii) $G$ cannot be covered by finitely many algebraic subsets, i.e., subsets of the form $\{x\in G:x{c}_{1}x{c}_2\cdots x{c}_n=e\}$ for some $c_1,c_2,\cdots ,c_n\in G$.

arXiv comment23 pages