Braid groups of the projective plane, mapping class groups of non-orientable surfaces and algebraic K-theory of their group rings

We describe the lower algebraic $K$-theory of the integral group ring of both the pure and full braid groups of the real projective plane $\mathbb{R}{P}^2$ with $3$ strings, as well as that of the integral group ring of the mapping class group of $\mathbb{R}{P}^2$ with $3$ marked points. In addition, we give a general formula for the algebraic $K$-theory groups of the group ring of the mapping class group of non-orientable surfaces with k marked points, where $k\ge 3$.