# On the Mod-2 Cohomology of SL 3 (z[ 1 2 , i])

## TL;DR

This paper computes the mod-2 cohomology of the group SL 3 over Z[1/2, i], revealing its structure in high degrees and supporting Quillen's conjecture about the cohomology ring.

## Contribution

It provides the first explicit calculation of the mod-2 cohomology for SL 3 over Z[1/2, i], confirming predictions about its structure in high degrees.

## Key findings

- Cohomology matches that of the group in degrees >8
- Supports Quillen's conjecture on cohomology ring structure
- Provides explicit cohomology calculations for a specific arithmetic group

## Abstract

Let $\Gamma$ = SL 3 (Z[ 1 2 , i]), let X be any mod-2 acyclic $\Gamma$-CW complex on which $\Gamma$ acts with finite stabilizers and let Xs be the 2-singular locus of X. We calculate the mod-2 cohomology of the Borel constructon of Xs with respect to the action of $\Gamma$. This cohomology coincides with the mod-2 cohomology of $\Gamma$ in cohomological degrees bigger than 8 and the result is compatible with a conjecture of Quillen which predicts the strucure of the cohomology ring H * ($\Gamma$; Z/2).