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 $\mathsf{T_{\!1}S}$-closed and projectively $\mathsf{T_{\!1}S}$-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 $\{x\in G:xc_1xc_2\cdots xc_n=e\}$ for some $c_1,c_2,\cdots,c_n\in G$.