Cohomology of SL(3,Z) with coefficients in the standard representation

TL;DR

This paper investigates the cohomology of SL(3,Z) with standard representation coefficients, revealing the existence of ghost classes and analyzing spectral sequence differentials to understand boundary cohomology behavior.

Contribution

It demonstrates that the second cohomology group H^2(SL(3,Z),V_3) contains ghost classes and links the non-triviality of spectral sequence differentials to these ghost classes.

Findings

H^2(SL_3(Z),V_3) has ghost classes.

The d_2 differential in the spectral sequence is non-trivial if ghost classes exist.

Spectral sequence related to GL_4(Z) does not degenerate at E_2.

Abstract

This paper is a natural continuation of a joint paper with Bajpai, Harder and Moya Giusti \cite{BHHM}, even though it began as an answer to Goncharov's question. It that paper, we had complete description for all representations except for odd symmetric powers and their dual ones. For those representations we were left with two options: certain one dimensional module is a ghost space or not. Here we find the $H^2(SL_3(\Z),V_3)$ has ghost classes. It means that it is generated by a class from the cohomology of the Borel subgroup. With the techniques developed here, we show that the $d_2$ map of the spectral sequence for the boundary cohomology of $GL_4(\Z)$ is non-trivial if and only if there is a ghost class in $GL_3(\Z)$ (see Propositions 11 and 12.) We use a result of Elbaz-Vincent, Gangl and Soule to show that a spectral sequence related to $GL_4(\Z)$ does not degenerate at $E_2$-level. Then $d_2$ is non-trivial. Therefore, we obtain that $H^2(SL_3(\Z),V_3))$ is a ghost space, where $V_3$ is the standard representation.