Mostow rigidity for skew solenoidal manifolds

TL;DR

This paper extends Mostow rigidity to foliated bundles and skew solenoidal manifolds over hyperbolic bases, demonstrating rigidity under certain invariant measures and holonomy actions.

Contribution

It proves a Mostow rigidity theorem for foliated bundles and skew solenoidal manifolds with invariant measures and twisted holonomy actions.

Findings

Rigidity holds for foliated bundles over hyperbolic manifolds with invariant measures.

Extends rigidity results to skew solenoidal manifolds with cocycle-twisted holonomy.

Includes solenoidal manifolds as inverse limits of finite coverings.

Abstract

We prove a Mostow rigidity theorem for foliated bundles over closed hyperbolic manifolds of dimension $n \geq 3$ endowed with a completely invariant measure of full support. These include solenoidal manifolds obtained as inverse limits of directed systems of finite coverings of closed hyperbolic manifolds. This theorem then extends to skew solenoidal manifolds for which the action of the holonomy group is twisted by means of a cocycle.