Homological stability for classical groups

Associated to every group with a weak spherical Tits system of rank n+1 with an appropriate rank n subgroup, we construct a relative spectral sequence involving group homology of Levi subgroups of both groups. Using the fact that such Levi subgroups frequently split as semidirect products of smaller groups, we prove homological stability results for unitary groups over division rings with infinite centre as well as for special linear and special orthogonal groups over infinite fields.

💡 Research Summary

The paper develops a unified method for proving homological stability of a wide class of classical groups by exploiting the geometry of weak spherical Tits systems. The authors start with a group G of rank n + 1 equipped with a weak spherical Tits building and a natural rank‑n subgroup H. From the inclusion H ↪ G they construct a relative spectral sequence whose E₂‑page is expressed as the homology of Levi subgroups L_q of the two groups with suitable coefficient modules. The crucial observation is that, in most cases of interest, each Levi subgroup splits as a semi‑direct product L ≅ K ⋉ A where K is a smaller classical group (often a general linear group) and A is a commutative unipotent group (e.g., an upper‑triangular matrix group).

This decomposition allows the E₂‑terms to factor into a product of two homology groups: one for K, for which classical homological stability results (Quillen, van der Kallen, Charney, etc.) are already known, and one for A, whose homology vanishes in low degrees because A is a nilpotent, abelian group. By carefully estimating the vanishing range of the A‑part and the stability range for K, the authors obtain explicit degree bounds f(n) such that the spectral sequence collapses for total degree ≤ f(n). Consequently they prove that the inclusion G_n → G_{n+1} induces an isomorphism on homology in all degrees i ≤ f(n).

The paper applies this framework to three families of groups:

1. Unitary groups U_n(D) over a division ring D with infinite centre. Here the relevant Levi subgroups are GL_k(D) ⋉ N_k(D), where N_k(D) is the additive group of strictly upper‑triangular matrices. Using the known stability for GL_k(D) and the trivial homology of N_k(D) in low degrees, they obtain stability for U_n(D) in degrees i ≤ n − 2.

2. Special linear groups SL_n(F) over an infinite field F. The Levi subgroups are isomorphic to GL_{n‑1}(F) × F^×. The factor GL_{n‑1}(F) contributes the usual stability, while the central torus F^× is abelian and contributes no low‑degree homology. The resulting stability range is i ≤ ⌊(n‑2)/2⌋, which improves on several earlier bounds.

3. Special orthogonal groups SO_n(F) over an infinite field. Their Levi subgroups decompose as O_{n‑1}(F) × F^×, and by iterating the same argument one obtains stability for SO_n(F) in degrees i ≤ n − 3.

The authors formulate a general theorem: if a sequence of groups {G_n} admits a weak spherical Tits system and each Levi subgroup L_n splits as K_n ⋉ A_n with K_n a smaller classical group and A_n an abelian unipotent group, then the homology groups H_i(G_n) stabilize for i below an explicit linear function of n. The proof proceeds in four steps: (i) construction of the relative spectral sequence from the Tits building, (ii) factorisation of the E₂‑page using the semi‑direct product structure, (iii) application of existing stability results to the K‑part and vanishing results to the A‑part, and (iv) dimension counting to guarantee collapse of the spectral sequence.

Beyond the main stability theorems, the paper discusses several consequences. It yields new proofs of stability for algebraic K‑theory of division rings with infinite centre, provides a systematic way to compute higher homology of unitary groups over such rings, and suggests that the explicit nature of the spectral sequence makes it amenable to computer‑assisted calculations. In summary, the work introduces a powerful, geometry‑driven spectral sequence technique that unifies and extends homological stability results for a broad spectrum of classical groups.