Cohomology of special unitary groups and congruence subgroups

TL;DR

This paper proves a homotopy invariance result for the first cohomology group of the special unitary group over polynomial rings, showing it is isomorphic to that of PGL_2(F).

Contribution

It establishes a natural isomorphism between the first cohomology groups of SU_3(F[t]) and PGL_2(F), revealing a new invariance property.

Findings

First cohomology of SU_3(F[t]) is isomorphic to that of PGL_2(F)

Homotopy invariance holds for these cohomology groups

Provides new insights into the structure of special unitary groups over polynomial rings

Abstract

We prove a homotopy invariance result for the first cohomology group of the special unitary group $\mathrm{SU}_3(F[t])$ with coefficients in irreducible representations of $\mathrm{PGL}_2(F)$. The main theorem establishes that this cohomology is naturally isomorphic to the corresponding cohomology of $\mathrm{PGL}_2(F)$.