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The cohomology of arithmetic groups is made up of two pieces, the cuspidal and noncuspidal parts. Within the cuspidal cohomology is a subspace— the  f -cuspidal cohomology—spanned by the classes that generate representations of the associated finite Lie group which are cuspidal in the sense of finite Lie group theory. Few concrete examples of  f -cuspidal cohomology have been computed geometrically, outside the cases of rational rank 1, or where the symmetric space has a Hermitian structure. This paper presents new computations of the  f -cuspidal cohomology of principal congruence subgroups      Γ  ( p ) \Gamma (p)       of      GL ( 3 , Z ) {\text {GL}}(3,\mathbb {Z})       of prime level  p . We show that the  f -cuspidal cohomology of      Γ  ( p ) \Gamma (p)       vanishes for all      p ⩽  19 p \leqslant 19       with      p ≠  11 p \ne 11      , but that it is nonzero for      p = 11 p = 11      . We give a precise description of the  f -cuspidal cohomology for      Γ  ( 11 ) \Gamma (11)       in terms of the  f -cuspidal representations of the finite Lie group      GL ( 3 , Z / 11 ) {\text {GL}}(3,\mathbb {Z}/11)      . We obtained the result, ultimately, by proving that a certain large complex matrix  M  is rank-deficient. Computation with the SVD algorithm gave strong evidence that  M  was rank-deficient; but to prove it, we mixed ideas from numerical analysis with exact computation in algebraic number fields and finite fields.

Doubly cuspidal cohomology for principal congruence subgroups of 𝐺𝐿(3,𝑍)