Narrow Results

In this paper we show that all finitely generated nilpotent, metabelian, polycyclic, and rigid (hence free solvable) groups $G$ are fully characterized in the class of all groups by the set $tp(G)$ of types realized in $G$. Furthermore, it turns out that these groups $G$ are fully characterized already by some particular rather restricted fragments of the types from $tp(G)$. In particular, every finitely generated nilpotent group is completely defined by its ${\exists }^{+}$-types, while a finitely generated rigid group is completely defined by its $\forall$-types, and a finitely generated metabelian or polycyclic group is completely defined by its $\forall \exists$-types. We have similar results for some non-solvable groups: free, surface, and free Burnside groups, though they mostly serve as illustrations of general techniques or provide some counterexamples.

We survey two of the many aspects of the standard braid order, namely its set theoretical roots, and the known connections with knot theory, including results by Netsvetaev, Malyutin, and Ito, and very recent work in progress by Fromentin and Gebhardt.