Congruence Subgroup Property for nilpotent groups and subsurface subgroups of Mapping Class Groups

TL;DR

This paper proves the Congruence Subgroup Property for specific subgroups of Mapping Class Groups, including those related to nilpotent quotients and subsurface inclusions, providing new elementary proofs and geometric insights.

Contribution

It establishes the Congruence Subgroup Property for two new families of subgroups of Mapping Class Groups, with elementary proofs and geometric interpretations.

Findings

Proved CSP for nilpotent quotient related subgroups.

Established CSP for subsurface inclusion subgroups.

Provided an elementary proof of a theorem by Ben-Ezra-Lubotzky.

Abstract

We prove the Congruence Subgroup Property for two families of subgroups of Mapping Class Groups of finite-type surfaces. The first one is related to nilpotent quotients of the fundamental group and Johnson filtration, and along the way we give an elementary proof of a theorem of Ben-Ezra-Lubotzky. The second family consists of certain geometric subgroups obtained via subsurface inclusions.