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

TL;DR

This paper introduces a new deformation method for representations of fundamental groups into Lie groups, extending earthquake deformations to higher-dimensional hyperbolic surface groups and their laminations.

Contribution

It develops a general deformation framework based on hypersurfaces and extends earthquake deformations to $SO(n,1)$ surface groups via measured laminations.

Findings

Deformations are defined using codimension 1 hypersurfaces.

Deformation operations commute when hypersurfaces are disjoint.

Extended earthquake deformations are achieved on $SO(n,1)$ 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 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 $SO(n,1)$ and show that the construction can be extended continuously to measured laminations, thus obtaining earthquake deformations on these surface groups.