Abstract:

Rigidity phenomena in negatively curved spaces are classically exemplified by Mostow rigidity. Later, Schwartz’s pattern rigidity theorems showed that coarse preservation of geometric patterns can already force rigidity, even without equivariance assumptions.

In this talk, I study pattern rigidity for Fuchsian groups in the spirit of Schwartz.

Although absolute continuity can recover Mostow-type rigidity, it is not sufficient for pattern rigidity of Fuchsian groups. Indeed, there are many counterexamples with non-chaotic behavior. Moreover, I construct chaotic counterexamples of higher regularity arising from the geometry of Cantor systems on the limit sets, revealing new forms of wild flexibility.

I also discuss several mechanisms that restore rigidity, including weak equivariance assumptions and geometric conditions associated with filling curves on surfaces.