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

Abstract

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.