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# Mathematical Research Letters

## Volume 21 (2014)

### Number 5

### Cohomology of $GL_4 (\mathbb{Z})$ with nontrivial coefficients

Pages: 1111 – 1136

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n5.a9

#### Author

#### Abstract

We compute the cohomology groups of $GL_4 (\mathbb{Z})$ with coefficients in symmetric powers of the standard representation twisted by the determinant. This problem arises in Goncharov’s approach to the study of motivic multiple zeta values of depth $4$. We use a result of Harder on Eisenstein cohomology and a computationally effective version for the homological Euler characteristic of arithmetic groups.