# Homological vanishing for the Steinberg representation

## TL;DR

This paper proves that certain homology groups of classical groups with coefficients in Steinberg representations vanish under specific conditions, advancing understanding of their algebraic and topological properties.

## Contribution

It establishes new homological vanishing results for classical groups with Steinberg coefficients, extending previous knowledge in algebraic topology and representation theory.

## Key findings

- Homology groups of classical groups vanish in specified degrees
- Vanishing occurs for groups like GL, SL, Sp, and SO under given conditions
- Results apply to a range of algebraic groups with Steinberg representations

## Abstract

For a field $k$, we prove that the $i$th homology of the groups $GL_n(k)$, $SL_n(k)$, $Sp_{2n}(k)$, $SO_{n,n}(k)$, and $SO_{n,n+1}(k)$ with coefficients in their Steinberg representations vanish for $n \geq 2i+2$.