Cohomology of congruence subgroups of SL3(Z), Steinberg modules, and real quadratic fields

We investigate the homology of a congruence subgroup Gamma of SL3(Z) with coefficients in the Steinberg modules St(Q3) and St(E3), where E is a real quadratic field and the coefficients are Q. By Borel-Serre duality, H0(Gamma,St(Q3)) is isomorphic to H3(Gamma, Q). Taking the image of the connecting homomorphism H1(Gamma,St(E3)/St(Q3)) -> H0(Gamma,St(Q3)), followed by the Borel-Serre isomorphism, we obtain a naturally defined Hecke-stable subspace H(Gamma, E) of H3(Gamma, Q). We conjecture that H(Gamma, E) is independent of E and consists of the cuspidal cohomology Husp(Gamma,Q) plus a certain subspace of H3(Gamma, Q) that is isomorphic to the sum of the cuspidal cohomologies of the maximal faces of the Borel-Serre boundary. We report on computer calculations of H(Gamma, E) for various Gamma, E which provide evidence for the conjecture. We give a partial heuristic for the conjecture.(c) 2022 Elsevier Inc. All rights reserved.