Quantitative Representation Stability Over Linear Groups

Abstract

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.

Representation Stability and Finite Orthogonal Groups

“…Similar Noetherian properties and asymptotic structure theorems were proven, as well as broad homological stability theorems with twisted coefficients. Some of these results were strengthened in [12], where an explicit bound for the stability degree was shown.…”

Effective and Infinite-Rank Superrigidity in the Context of Representation Stability

“…For V IC(F q )-modules Miller and Wilson have obtained some very tight bounds for these sort of questions (in [14]), it is likely that some of their analysis can be carried over to this setting as well.…”

“…V IC(R)-modules for R a commutative ring were defined by Putman and Sam in [17], they are analogs of F I-modules with general linear groups GL n (R) playing the same role that symmetric groups played for F I-modules (see Section 3 for a precise definition). The most well-understood case is when R = F q is a finite field (studied in [17], [10], and [14]), where we now know many of the same properties as for F I-modules.…”

Representation stability, secondary stability, and polynomial functors

“…Equivalently, a stability category U G is degree-wise coherent if the presentation degree of a U G-module can be used to find finite bounds for the generation degree of higher syzygies (see e.g. [MW,Corollary 2.36]). The following definition quantifies this.…”