(Non-)Vanishing of high-dimensional group cohomology

TL;DR

This survey analyzes high dimensional rational cohomology for duality groups akin to SL_n(Z), including MCG and Aut(F_n). It leverages Borel–Serre duality and partial resolutions of Steinberg modules to explain vanishing in low codimensions and surveys non-vanishing phenomena across rings and group types, with detailed treatments of SL_n, Sp_{2n}, Chevalley groups, and congruence subgroups. Key contributions include clarifying the ranges where vanishing holds (notably in Euclidean settings) and presenting clear counterexamples and growth results that show stability breaks down in many cases, while also outlining explicit partial resolutions and presentations of Steinberg modules. The work maps the landscape of high-dimensional cohomology, identifying where conjectures such as CFP and Brueck’s vanishing may still hold and where fundamentally new phenomena arise, guiding future research on dualising modules and arithmetic group cohomology. The results have implications for understanding the cohomology of arithmetic groups, moduli spaces, and configuration-space related structures, and they underscore the subtle dependence on ring structure and group type in high-dimensional stability patterns.

Abstract

Church-Farb-Putman formulated stability and vanishing conjectures for the high-dimensional cohomology of $\operatorname{SL}_n(\mathbb{Z})$, surface mapping class groups and automorphism groups of free groups. This is a survey on the current status of these conjectures and their generalisations.

Figures (5)

• Figure 1: $H^k(\operatorname{SL}_{n}(\mathbb{Z});\mathbb{Q})$. The pink area ${n \choose 2} - (n-2)\leq k \leq {n\choose 2}$ is the range of \ref{['conj_cfp']}. Triangle, square and circle mark existing vanishing results (\ref{['eq_lee_szcarba']}, \ref{['eq_church_putman']} and \ref{['eq_bpmsw']}); brown the known non-trivial classes of highest degree (\ref{['eq_brown']} for $n\geq 8$).
• Figure 2: An apartment in $\Delta(\operatorname{SL}_{3}(R))$, corresponding to a basis $\vec{v}_1, \vec{v}_2, \vec{v}_3$ of $\mathbb{K}^{3}$.
• Figure 3: A 2-simplex in $\operatorname{BA}_2$ whose boundary leads to Relation 2. in $\operatorname{St}(\operatorname{SL}_{2}(\mathbb{Z}))$. Its 1-faces correspond to apartment classes $[v_1, v_2]$, $[v_1, \langle \vec{v}_1+\vec{v}_{2} \rangle]$, $[\langle \vec{v}_1+\vec{v}_{2} \rangle, v_2]$ and are contained in $\operatorname{B}_2$.
• Figure 4: Number rings under the assumption of GRH.
• Figure 5: A subcomplex of $\operatorname{IAA}_2$ whose boundary leads to Relation 3. in $\operatorname{St}(\operatorname{Sp}_{4}(\mathbb{Z}))$.

Theorems & Definitions (2)

• Conjecture 1: Church--Farb--Putman
• Conjecture 2