On the geometry of some braid group representations (Q2329227)

This paper is a ``conference paper'' where the author makes a survey developing earlier results. More precisely, firstly, a low-dimensional matrix representations of pure braid groups is presented obtained via homology of suitable nilpotent flat connections appeared in [\textit{A. Benvegnù} and the author, ``Low-dimensional pure braid group representations via nilpotent flat connections'', Boll. Unione Mat. Ital. (9) 6, No. 3, 643--672 (2013)]. Here the author gives a coarse and short introduction to standard and pure braid groups and discusses \(P_{3}\) and \(P_{4}\) representations referring, for details, to the above-mentioned paper. Subsequently, after a brief motivating physical background, a review of the construction of unitary representations of Riemann surface braid groups is given. The whole development is based on [the author, J. Geom. Phys. 94, 120--140 (2015; Zbl 1317.81178)] and is summarized in two theorems (Theorems 4 and 5) where for a more detailed study the interested reader is referred to the above paper. The paper concludes with some thoughts for possible research directions. For the entire collection see [Zbl 1419.57001].