Central stability homology

We introduce a technique for proving quantitative representation stability theorems for sequences of representations of certain finite linear groups over a field of characteristic zero. In particular, we prove a vanishing result for higher syzygies of VIC and SI-modules, which can be thought of as a weaker version of a regularity theorem of Church-Ellenberg [CE17, Theorem A] in the context of FI-modules. We apply these techniques to the rational homology of congruence subgroups of mapping class groups and congruence subgroups of automorphism groups of free groups.This partially resolves a question raised by Church and Putman-Sam [PS17, Remark 1.8]. We also prove new homological stability results for mapping class groups and automorphism groups of free groups with twisted coefficients.Jeremy Miller was supported in part by National Science Foundation grant DMS-1709726.1.1. Stability for congruence subgroups. The study of representation stability concerns the following framework: fix a sequence of groups with inclusionssuch as symmetric groups S n , general linear groups GL n (k), or symplectic groups Sp 2n (k). Fix a commutative ring R. Let {A n } be a sequence of R[G n ]-modules with the data of G n -equivariant maps A n → A n+1 . The sequence {A n } is said to have generation degree ≤ d if, for all n ≥ d, the R[G n+1 ]-module generated by the image of A n is all of A n+1 . Informally, we say that the sequence {A n } stabilizes if its generation degree is finite. In this paper we also discuss a related notion called presentation degree.The main examples of spaces that we consider are classifying spaces of congruence subgroups of mapping class groups and congruence subgroups of automorphism groups of free groups. Let Mod(Σ g,r ) denote the mapping class group of Σ g,r , the compact orientable surface of genus g with r boundary components. The mapping class group acts on H 1 (Σ g,r ). For r ≤ 1, this action preserves the symplectic intersection form and so we get a map Mod(Σ g,r ) → Sp 2g (Z) to the group Sp 2g (Z) of symplectomorphisms of Z 2g . Reducing modulo p gives a map Mod(Σ g,r ) → Sp 2g (Z/pZ) and we denote the kernel by Mod(Σ g,r , p). This group is often called the level p congruence subgroup of Mod(Σ g,r ). For r = 0, the classifying space of this group has the homotopy type of the moduli stack of smooth genus g complex curves with full level p structure. For r ≤ 1, the homology groups H i (Mod(Σ g,r , p); R) have the structure of a R[Sp 2g (Z/pZ)]-module. For r = 1, the inclusions of surfaces Σ g,1 ֒→ Σ g+1,1 induce Sp 2g (Z/pZ)-equivariant maps H i (Mod(Σ g,1 , p); R) → H i (Mod(Σ g+1,1 , p); R)which allow us to make sense of stability. Our first result is the following.

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“…It is a slight sharpening of the above theorem for the induced module M(0). Proposition 3.3 (Patzt [Pat17a,Remark 5.6]). Let k be a field.…”

Section: Representation Stability Resultsmentioning

confidence: 99%

Quantitative Representation Stability Over Linear Groups

Miller¹,

Wilson

2018

International Mathematics Research Notices

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“…In Section 3.1, we recall the definition of the categories VIC H (R), SI(R), and FI. Our approach to these topics uses the formalism of stability categories developed by the second author [Pat17]. We Definition 3.1.…”

Section: Modules Over Stability Categoriesmentioning

confidence: 99%

“…For all categories considered in this paper, central stability is equivalent to presentability in finite degree. The statement concerning FI-modules is due to Church-Ellenberg [CE15] and the statement involving other categories is due to the second author [Pat17]. Note that the central stability degree of a U G-module is always at least its degree of generation.…”

Section: 2mentioning

confidence: 99%

“…FI-homology has appeared in work of Church, Ellenberg, Farb, and Nagpal "cyclically generated" complemented categories and used the notation Σ * +1 (A). In this paper, we adopt notation used by the second author [Pat17].…”

Section: 2mentioning

confidence: 99%

“…The main technical lemma of this paper is a result showing higher central stability for modules over certain categories with finite polynomial degree, such as modules over the categories appearing in work of Putman and Sam [PS14]. The proof of this lemma involves verifying a general acyclicity criterion developed by the second author [Pat17] for complexes associated to these modules. From this result we deduce that if a module satisfies a polynomial condition -a condition often appearing in twisted homological stability theorems -then this module exhibits a form of representation stability.…”

Section: Introductionmentioning

confidence: 99%

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Central stability for the homology of congruence subgroups and the second homology of Torelli groups

Miller¹,

Patzt²,

Wilson

2019

Advances in Mathematics

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We prove a representation stability result for the second homology groups of Torelli subgroups of mapping class groups and automorphism groups of free groups. This strengthens the results of Boldsen-Hauge Dollerup and Day-Putman. We also prove a new representation stability result for the homology of certain congruence subgroups, partially improving upon the work of Putman-Sam. These results follow from a general theorem on syzygies of certain modules with finite polynomial degree.D'] generalized both theorems with a result for integral homology with explicit stable ranges.Putman [Put15, fifth Remark] comments that it would be ideal to understand stability properties of GL n (R, I) as GL U n (R/I)-representations instead of just S n -representations. We provide the following partial solution to this problem.Theorem C. Let I be a two-sided ideal of a ring R and let t be the minimal stable rank of all rings containing I as a two-sided ideal. If R/I is a PID of stable rank s, then the central stability degree of the sequence H i (GL n (R, I)) isas GL U n (R/I)-representations.Remark 1.2. Note that the quotient R/I need not be a PID in order to define an action on H * (GL n (R, I)) by a group GL n (Q) with Q a PID. For example, given an non-unital ring I, wecan realize I as a two-sided ideal in its unitilization I + , its image under the left adjoint to the forgetful functor from the category of (unital) rings to the category of non-unital rings. Then I + /I ∼ = Z, and GL n (I + , I) ∼ = GL n (R, I). Hence in particular H * (GL n (R, I)) is always a GL n (Z)-representation.

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Representation stability for diagram algebras

Patzt

2024

Journal of Algebra

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Central stability homology

Cited by 15 publications

References 15 publications

Quantitative Representation Stability Over Linear Groups

Quantitative Representation Stability Over Linear Groups

Central stability for the homology of congruence subgroups and the second homology of Torelli groups

Representation stability for diagram algebras

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