A note on the rational homological dimension of lattices in positive characteristic

TL;DR

This paper demonstrates that for lattices in products of simple Chevalley groups over function fields, the rational homological dimension equals the cohomological dimension and matches the dimension of the associated Bruhat--Tits building, using $\, ext{ extltilde} ext{}^2$-homology.

Contribution

It establishes a precise equality between the rational homological dimension, cohomological dimension, and the Bruhat--Tits building dimension for these lattices, advancing understanding in algebraic and geometric group theory.

Findings

Rational homological dimension equals cohomological dimension.

Both dimensions match the Bruhat--Tits building dimension.

Uses $\, ext{ extltilde} ext{}^2$-homology techniques.

Abstract

We show via $\ell^2$-homology that the rational homological dimension of a lattice in a product of simple simply connected Chevalley groups over global function fields is equal to the rational cohomological dimension and to the dimension of the associated Bruhat--Tits building.