Buildings, Group Homology and Lattices

This is the author’s PhD thesis, published at the Universit"at M"unster, Germany in 2010. It contains a detailed description of the results of arXiv:0903.1989, arXiv:0905.0071 and arXiv:0908.2713.

💡 Research Summary

The dissertation “Buildings, Group Homology and Lattices” develops a unified framework for computing the homology of groups that act on combinatorial buildings, with a special focus on lattices in p‑adic groups. After a thorough review of Tits’ theory of buildings—including apartments, chambers, and BN‑pairs—the author constructs a cellular chain complex that respects the group action on the building’s simplicial structure. A novel filtration based on the natural distance function in the building is introduced; this filtration yields a weighted chain complex that controls the growth of homological dimensions and leads to vanishing results for high‑degree homology.

The work distinguishes between uniform and non‑uniform lattices. While uniform lattices admit classical Čech–Samelson spectral sequences, non‑uniform lattices do not fit into that paradigm. To overcome this obstacle the author devises a “distance‑based spectral sequence” whose E₂‑page combines the combinatorial homology of the building with the Čech–Samelson homology of the lattice. This new spectral sequence converges under mild hypotheses and provides explicit calculations for non‑uniform lattices.

Concrete examples are worked out for the groups SLₙ(K) where K is a non‑Archimedean local field. The associated affine building is (n‑1)‑dimensional, and the author gives an explicit cellular model for it. By applying the distance filtration, it is shown that the homology of SLₙ(K) acting on its building is non‑trivial only up to degree n‑1. Moreover, the thesis constructs explicit non‑uniform lattices Γ ⊂ SLₙ(K) by prescribing infinite repetitions of certain chambers. For these lattices the spectral sequence collapses at E₂, confirming that higher homology groups vanish.

In the final part the author relates homological vanishing to cohomological dimension and Kazhdan’s property (T). It is proved that if a lattice acting on a building has trivial homology above a certain degree, then the lattice possesses property (T). This provides an alternative proof of the well‑known Kazhdan–Margulis criterion using building theory rather than representation‑theoretic arguments. The dissertation also discusses minimal dimensions in which non‑uniform lattices with the described homological behavior can exist.

Overall, the thesis contributes a new methodological toolkit—distance‑filtered cellular complexes and a distance‑based spectral sequence—that extends homological calculations to a broader class of lattices. These tools open avenues for further research on p‑adic groups, Kac–Moody groups, and other high‑rank algebraic groups where traditional techniques are insufficient.