$o$-bounded groups and other topological groups with strong combinatorial properties

We construct several topological groups with very strong combinatorial properties. In particular, we give simple examples of subgroups of $\mathbb{R}$(thus strictly $o$-bounded) which have the Menger and Hurewicz properties but are not $\sigma$-compact, and show that the product of two $o$-bounded subgroups of $\mathbb{R} ^{\mathbb{N} }$ may fail to be $o$-bounded, even when they satisfy the stronger property $\mathsf{S}_1(\mathcal{B}_{\Omega},\mathcal{B}_{\Omega})$. This solves a problem of Tkacenko and Hernandez, and extends independent solutions of Krawczyk and Michalewski and of Banakh, Nickolas, and Sanchis. We also construct separable metrizable groups $G$ of size continuum such that every countable Borel $\omega$-cover of $G$ contains a $\gamma$-cover of $G$.