On the non-realizability of braid groups by diffeomorphisms
with N. Salter
Bull. Lond. Math. Soc., 48 (2016) 3, 457–471.
For every compact surface we show that when n is sufficiently large there is no lift of the surface braid group Br(n,S) to Diff(S,n), the group of diffeomorphisms preserving n marked points and restricting to the identity on the boundary. This generalizes work of Bestvina-Church-Souto, but our main tool (the Thurston stability theorem) is different. Our methods are applied to give a new proof of Morita’s non-lifting theorem in the best possible range. These techniques extend to the more general setting of spaces of codimension-2 embeddings, and we obtain corresponding results for spherical motion groups, including the string motion group.