Uniform twisted homological stability

Technical Summary: Uniform Twisted Homological Stability

Problem Statement
The paper addresses the problem of homological stability for families of discrete groups (such as mapping class groups, automorphism groups of free groups, and braid groups) with coefficients in sequences of irreducible algebraic representations of arithmetic groups. While homological stability for these groups with constant or polynomial coefficients is well-established, the "stable range" (the degree $d$ up to which stability holds) typically depends on the degree of polynomiality of the coefficient system. Specifically, for a coefficient system of polynomial degree $r$, the stable range is often of the form $d < A \cdot n - B \cdot r$. Consequently, if the degree of the representation grows with $n$, the stable range may vanish entirely.

The authors aim to prove a "uniform" stability theorem where the stable range depends on the group index $n$ but is independent of the degree of the polynomiality of the coefficient system, provided the coefficients are specific irreducible algebraic representations.

Methodology
The paper develops a general framework for propagating homological stability from a family of arithmetic groups $\{Q_n\}$ to a family of discrete groups $\{\Gamma_n\}$ that map to them.

1. General Propagation Theorem (Section 2): The authors formulate Theorem 2.2, which establishes conditions under which stability for $\{Q_n\}$ implies stability for $\{\Gamma_n\}$. The theorem relies on three axioms:

   • Axiom (I): A connectivity condition on the "destabilisation complexes" (or splitting complexes) associated with the families $\{\Gamma_n\}$ and $\{Q_n\}$. This involves the derived indecomposables of the associated $E_2$-algebras.
   • Axiom (II): A stability condition for the coefficient system $V$ over the arithmetic groups $\{Q_n\}$, specifically regarding the vanishing of relative homology $H_*(Q_n, Q_{n-1}; \Sigma^a V)$.
   • The Propagation Mechanism: Using the language of braided monoidal groupoids and $E_2$-algebras (following the work of Randal-Williams and Wahl), the authors construct a spectral sequence relating the homology of $\Gamma_n$ to that of $Q_n$ and the kernels of the maps $\Gamma_n \to Q_n$.

2. Verification of Axioms for Specific Cases (Section 4):

   • Axiom (I): The authors verify high connectivity of the relevant destabilisation complexes. For mapping class groups and free group automorphisms, they rely on existing results (e.g., by Hatcher, Wahl, and others). For the braid groups (Theorem 1.4), they prove new high-connectivity results for the complexes associated with the image of the integral Burau representation in even and odd symplectic groups (Section 3).
   • Axiom (II): This requires analyzing the behavior of irreducible representations under the shift operator $\Sigma$ (restriction along $Q_{n-1} \to Q_n$). The authors utilize Littlewood complexes (Sam–Snowden–Weyman) to decompose shifted representations into iterated extensions of other irreducible representations. This allows them to reduce the stability of shifted coefficients to the stability of the base coefficients, leveraging known results on the stable cohomology of arithmetic groups (Borel's work).

3. Destabilisation Complexes for Burau Representation (Section 3): A significant technical contribution is the construction and analysis of the destabilisation complexes for the image of the integral Burau representation in the "odd symplectic groups" (Gelfand–Zelevinsky). The authors prove these complexes are highly connected (specifically, $(n-19)/8$-connected), which is necessary to establish the slope of stability for braid groups.

Key Contributions and Results
The paper proves four main theorems establishing uniform twisted homological stability:

1. Mapping Class Groups (Theorem 1.1): For the mapping class group $\text{Mod}^1_g$ with coefficients in an irreducible representation $V_\lambda(2g)$ of $\text{Sp}_{2g}(\mathbb{Z})$, the relative homology vanishes for $d < \frac{1}{2}(g - \frac{1}{2}l(\lambda))$. This improves the dependence on the partition length $l(\lambda)$ compared to previous bounds.
2. Automorphism Groups of Free Groups (Theorem 1.2): For $\text{Aut}(F_n)$ with coefficients in irreducible representations $W_{\lambda\mu}(n)$ of $\text{GL}_n(\mathbb{Z})$, stability holds for $d < \frac{1}{2}(n - \beta)$, where $\beta$ depends on the lengths and conjugate lengths of the partitions. The authors clarify that this constitutes a uniform stability of slope $1/4$ because for any irreducible representation, there exists a choice of $(\lambda, \mu)$ such that $\beta \leq \frac{1}{2}(n+2)$.
3. Handlebody Mapping Class Groups (Theorem 1.3): Analogous uniform stability is proven for handlebody mapping class groups with coefficients pulled back from $\text{GL}_n(\mathbb{Z})$.
4. Braid Groups (Theorem 1.4): For the braid group $\beta_n$ with coefficients in representations of the image of the Burau representation (odd/even symplectic groups), stability holds for $d < \frac{1}{17}(n - \max(2, l(\lambda)))$.

Significance and Applications

• Uniformity: The primary novelty is that the stable range does not depend on the degree of polynomiality of the coefficient system. This allows for stability results in regimes where the degree of the representation grows with $n$ (e.g., $|\lambda| \gg n$), a situation where previous bounds yielded no stable range.
• Arithmetic Statistics: The authors demonstrate a direct application of Theorem 1.4 to the moments of families of quadratic $L$-functions over function fields. Combined with prior work by Bergström–Diaconu–Petersen–Westland, the new stability range proves the Conrey–Farmer–Keating–Rubinstein–Snaith (CFKRS) predictions for all moments of these $L$-functions for sufficiently large odd prime powers. This is the first nontrivial family where CFKRS asymptotics are established for all moments.
• Torelli Groups: The results provide tools for studying the homology of Torelli groups (kernels of the maps to arithmetic groups). Specifically, Theorem 1.1 implies that the homology of the Torelli group with twisted coefficients can be calculated in any degree for sufficiently large genus, potentially simplifying proofs regarding quadratic presentations of the Malcev completion of Torelli groups.
• Representation Stability: The work provides evidence for "uniform representation stability" of the homology of Torelli subgroups, extending the theory of representation stability (Church–Farb) to these specific contexts.

Limitations and Sharpness
The authors note that their stable ranges are likely not optimal. They provide upper bounds on the possible slopes of stability based on known stable homology calculations (e.g., by Madsen–Weiss, Galatius, and Lindell). For instance, they conjecture that the slope $1/4$ for mapping class groups and handlebody groups is sharp, while the slope $1/17$ for braid groups is likely not optimal (with a conjectured bound of $1/3$). The dependence on the partition length in the braid group case is also noted as potentially improvable.