Deformations of fundamental group representations and earthquakes on $${textit{SO}}(n,1)$$ surface groups

In this article we construct a type of deformations of representations (pi _1(M)rightarrow G) where G is an arbitrary lie group and M is a large class of manifolds including (hbox {CAT}(0)) manifolds. The deformations are defined based on codimension 1 hypersurfaces with certain conditions, and also on disjoint union of such hypersurfaces, i.e. multi-hypersurfaces. We show commutativity of deforming along disjoint hypersurfaces. As application, we consider Anosov surface groups in ({textit{SO}}(n,1)) and show that the construction can be extended continuously to measured laminations, thus obtaining earthquake deformations on these surface groups.