The congruence subgroup property for mapping class groups and the residual finiteness of hyperbolic groups

TL;DR

This paper proves the congruence subgroup property for mapping class groups of hyperbolic surfaces and shows that hyperbolic 3-manifolds with the same profinite completions are commensurable, assuming hyperbolic groups are residually finite.

Contribution

It establishes the congruence subgroup property for mapping class groups under the residual finiteness assumption for hyperbolic groups and links profinite equivalence to commensurability of hyperbolic 3-manifolds.

Findings

Proves the congruence subgroup property for mapping class groups of hyperbolic surfaces.

Shows profinitely equivalent hyperbolic 3-manifolds are commensurable.

Assumes all hyperbolic groups are residually finite.

Abstract

Assuming that every hyperbolic group is residually finite, we prove the congruence subgroup property for mapping class groups of hyperbolic surfaces of finite type. Under the same assumption, it follows that profinitely equivalent hyperbolic 3-manifolds are commensurable.