Quasi-isometric Rigidity

Description: The notion of a quasi-isometry is central to geometric group theory: it asks whether two groups/spaces are “coarsely isometric” to each other. It turns out that a lot of properties are invariant under quasi-isometries, such as hyperbolicity.

Quasi-isometric (QI) rigidity is a property of groups that intuitively says that everything quasi-isometric to our group is “very similar” to it. More concretely, a group is QI rigid if any group/space that is quasi-isometric to it is actually virtually isomorphic to it. Examples of QI rigid classes of finitely generated groups are hyperbolic groups, free groups, groups with solvable word problem and more!

We will be following these notes on QI rigidity by Michael Kapovich.

We will also study Mostow’s rigidity theorem, which is another very interesting rigidity result. It regards hyperbolic manifolds of dimension at least 3, and asserts that their geometry is determined by their fundamental group! If there exists an isomorphism between the fundamental groups of two (connected, compact, oriented) hyperbolic manifolds, then this induces an isometry between them. This actually works by finding a quasi-isometry between the universal covers.

We will study these notes on Mostow’s rigidity theorem by José Andrés Rodriguez Migueles.