In recent years there has been quite some interest in the ``simplest'' solutions of the Yang-Baxter equation. Such solutions are involutive bijective mappings $r:X\times X \rightarrow X\times X$, where $X$ is a finite set, so that $r_{1}r_{2}r_{1}=r_{2}r_{1}r_{2}$, with $r_{1}=r\times id_{X}$ and $r_{2}=id_{X} \times r$. In case $r$ satisfies some non-degeneracy condition, Gateva-Ivanova and Van den Bergh, and also Etingof, Schedler and Soloviev, gave a beautiful group (monoid) theoretical interpretation of such solutions. Such groups (monoids) are said to be of $I$-type. In this lecture we give a survey of recent results on the algebraic structure of these groups (monoids) and their group (monoid) algebras.

A $G$-cover of a smooth projective curve $X$ over some algebraically closed field of characteristic zero ramified at a finite set $D$ of points, may be identified with a tensor functor from the category of finite representations of $G$ into bundles with parabolic structure along $D$, by work of Nori. Now when $X$ is the sphere and $D= \{0,1,\infty\},$ the parabolic bundle is of the form of $\oplus\mathcal{O}(s_i)$ for some integers $s_i$. These constants are very difficult to determine in general, but Ajneet Dhillon devised a clever method of bounding them using group theoretic data, and this is the subject of my talk.