The Congruence Subgroup Property for Subsurface Subgroups of Mapping Class Groups and Nilpotent Quotients

Core Concepts

This article investigates the Congruence Subgroup Property (CSP) in the context of mapping class groups of surfaces, demonstrating that CSP holds for significant families of subgroups related to nilpotent quotients and subsurface inclusions.

Abstract

• Bibliographic Information: Klukowski, A. (2024). Congruence Subgroup Property for Nilpotent Groups and Subsurface Subgroups of Mapping Class Groups. arXiv preprint arXiv:2411.06867v1.

• Research Objective: This paper aims to address the open question of whether the Congruence Subgroup Property (CSP) holds for mapping class groups of surfaces with genus at least 3. The author investigates this by examining CSP for specific families of subgroups within mapping class groups.

• Methodology: The author utilizes a divide-and-conquer approach, leveraging a composition lemma for CSP to break down the problem. The paper delves into the algebraic structure of mapping class groups, employing tools like the Andreadakis-Johnson filtration and Johnson homomorphisms to analyze their subgroups. Specific techniques include constructing special finite quotients of nilpotent groups and utilizing virtual retractions to relate quotients of subsurface fundamental groups to the whole surface.

• Key Findings:

  • The paper proves that for finitely generated nilpotent groups, the CSP of the outer automorphism group can be determined by analyzing the action on its abelianization (Theorem 4).
  • It demonstrates that the CSP holds for subgroups of mapping class groups that contain a term of the Johnson filtration (Corollary 5).
  • The research establishes that the CSP is preserved under gluing surfaces along boundaries, provided the subgroups on the subsurfaces satisfy CSP (Theorem 7).
  • As a consequence of these findings, the paper proves CSP for handle-pushing subgroups, stabilizers of large simplices in the curve complex, and, conditionally, for stabilizers of non-separating curves in mapping class groups (Corollaries 8, 9, 11).

• Main Conclusions: This work provides significant progress towards understanding the Congruence Subgroup Property in the context of mapping class groups. While the general question of CSP for mapping class groups of genus at least 3 remains open, the results presented offer valuable insights and tools for further investigation. The author suggests that proving a specific conjecture (Conjecture 12) regarding the containment of orbits of curves under congruence subgroups would be sufficient to establish the general CSP for mapping class groups.

• Significance: Understanding the finite-index subgroups of mapping class groups, particularly through the lens of CSP, has profound implications for various areas of topology and group theory. This includes problems related to profinite rigidity of 3-manifold groups, the Ivanov conjecture on virtual fibrations of moduli spaces, and the study of Property (T) in mapping class groups.

• Limitations and Future Research: The paper primarily focuses on specific families of subgroups within mapping class groups. The general question of CSP for all finite-index subgroups remains open and poses a significant challenge for future research. The author's Conjecture 12, if proven, would bridge this gap and establish the general CSP for mapping class groups. Further investigation into the algebraic structure of mapping class groups and the development of new techniques for analyzing their subgroups are crucial for addressing this open problem.

Stats

Quotes

"The question of CSP in genus at least 3 is open, and mentioned in the Kirby List [Kir97, Problem 2.10]."
"One motivation for this question is its link to the study of profinite rigidity of 3-manifold groups."
"Usually, CSP is thought of as a property of the automorphism group. However, our approach is to look at analogous statements about simpler subgroups, similar to the philosophy of [McR10] dealing with point-pushing subgroups and [Bog24] investigating centralisers of finite subgroups."

Key Insights Distilled From

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

by Adam Klukows... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.06867.pdf

Deeper Inquiries

How might the techniques and results presented in this paper be extended or adapted to study other properties of mapping class groups beyond the Congruence Subgroup Property?

This paper presents a powerful strategy for tackling the Congruence Subgroup Property (CSP) in mapping class groups by dissecting the problem into smaller, more manageable pieces. This "divide and conquer" approach, coupled with the insightful use of nilpotent quotients and subsurface structures, opens up several avenues for investigating other properties of mapping class groups:

Property (T) and other rigidity properties: The paper establishes a connection between CSP and the Johnson filtration, which plays a crucial role in understanding the rigidity properties of mapping class groups. The techniques employed, particularly the analysis of the action of mapping class groups on nilpotent quotients, could be further developed to study Property (T) and other rigidity phenomena. For instance, one could investigate how the presence or absence of Property (T) for certain subgroups of the mapping class group influences the CSP of the entire group.

Subgroup structure and classification: The paper focuses on specific geometrically significant subgroups, such as handle-pushing subgroups and curve stabilizers. The methods used to analyze these subgroups, particularly the interplay between algebraic and geometric arguments, could be extended to study other important subgroups, such as those arising from surface bundles or Lefschetz fibrations. This could lead to a more comprehensive understanding of the subgroup structure and classification of mapping class groups.

Profinite properties and connections to 3-manifolds: The paper highlights the link between CSP and the profinite rigidity of 3-manifold groups. The techniques developed, especially the use of virtual retractions and the analysis of boundary twists, could be further exploited to explore other profinite properties of mapping class groups and their connections to 3-manifold topology. For example, one could investigate the relationship between the profinite completion of a mapping class group and the profinite completion of the fundamental group of the corresponding mapping torus.

Generalizations to other groups: While the paper focuses on mapping class groups, the underlying ideas and techniques, such as the composition lemma for CSP and the analysis of automorphisms of nilpotent groups, could potentially be adapted to study similar properties in other contexts, such as automorphism groups of free groups, braid groups, or other groups acting on hyperbolic spaces.

Could there be a counterexample to the Congruence Subgroup Property in mapping class groups of high genus, perhaps arising from a deeper connection to the geometry or topology of the underlying surfaces?

While the paper makes significant progress towards proving the Congruence Subgroup Property (CSP) for mapping class groups, the possibility of a counterexample, particularly in high genus, cannot be ruled out. Here are some potential sources of such counterexamples:

Geometric group theory perspective: Mapping class groups exhibit a rich interplay between algebraic and geometric structures. A potential counterexample to CSP might arise from a deeper understanding of this interplay, perhaps by exploiting geometric constructions that produce finite-index subgroups with unexpected algebraic properties. For instance, one could investigate subgroups arising from highly symmetric surfaces or those with specific geometric constraints.

Connections to Teichmüller theory: Mapping class groups act on Teichmüller space, which parametrizes marked hyperbolic structures on a surface. A potential counterexample to CSP might emerge from a deeper understanding of this action and its connection to the geometry of Teichmüller space. For example, one could investigate subgroups associated with specific regions or strata within Teichmüller space.

Analogies with other groups: While the analogy with arithmetic groups like SL(n,Z) initially suggests the plausibility of CSP for mapping class groups, there are also notable differences. For instance, mapping class groups are not arithmetic for genus greater than 2. Exploring these differences and drawing comparisons with other groups where CSP fails, such as SL(2,Z), might provide insights into potential counterexamples.

New topological constructions: A counterexample to CSP might arise from novel topological constructions on surfaces, leading to finite-index subgroups with unexpected properties. For instance, one could investigate subgroups associated with intricate curve systems, handlebody decompositions, or other topological structures on the surface.

What are the implications of the potential relationship between the Congruence Subgroup Property and the profinite rigidity of 3-manifold groups for our understanding of the geometrization of 3-manifolds?

The potential relationship between the Congruence Subgroup Property (CSP) and the profinite rigidity of 3-manifold groups has profound implications for our understanding of the geometrization of 3-manifolds:

Profinite invariants and geometric classification: Profinite rigidity implies that the profinite completion of the fundamental group of a hyperbolic 3-manifold determines the manifold up to commensurability. If CSP holds for mapping class groups, it would provide a powerful tool for studying profinite rigidity by relating the profinite completion of a 3-manifold group to the profinite completion of a corresponding mapping class group. This could lead to new methods for distinguishing 3-manifolds and understanding their geometric structures using profinite invariants.

Virtual properties and geometric decomposition: CSP for mapping class groups would have implications for the virtual properties of 3-manifold groups. For instance, it could be used to study virtually special groups, which are fundamental groups of 3-manifolds admitting a geometric decomposition into "nice" pieces. This could shed light on the relationship between the algebraic structure of 3-manifold groups and the geometric decompositions of the corresponding manifolds.

New approaches to open problems: The connection between CSP and profinite rigidity could potentially lead to new approaches for tackling open problems in 3-manifold topology, such as the Virtual Haken Conjecture or the Ehrenpreis Conjecture. For example, understanding the profinite properties of mapping class groups might provide insights into the virtual properties of 3-manifold groups, potentially leading to progress on these conjectures.

Deeper understanding of geometrization: The Geometrization Conjecture, now a theorem, states that every 3-manifold can be decomposed into pieces admitting geometric structures. The potential relationship between CSP and profinite rigidity suggests a deeper connection between the algebraic and geometric aspects of 3-manifolds. Understanding this connection could lead to a more comprehensive and unified understanding of the geometrization of 3-manifolds.