We give a variation on the proof of Mostow's rigidity theorem, for certain
hyperbolic 3-manifolds. This is based on a rigidity theorem for conjugacies between
piecewise-conformal expanding Markov maps. The conjugacy rigidity theorem is deduced
from a Livsic cocycle rigidity theorem that we prove for smooth, compact Lie
group-valued cocycles over piecewise smooth expanding Markov maps.
AbstractWe formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a {{\mathbb{Z}}}-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley’s parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes.
We give a characterization of isomorphisms between Schreier graphs in terms of the groups, subgroups and generating systems. This characterization may be thought as a graph analog of Mostow’s rigidity theorem for hyperbolic manifolds. This allows us to give a transitivity criterion for Schreier graphs. Finally, we show that Tarski monsters satisfy a strong simplicity criterion. This gives a partial answer to a question of Benjamini and Duminil-Copin.
Profinite rigidity for twisted Alexander polynomials
