Unknown top-degree vanishing for SL_n over non-Euclidean PID number rings

Ascertain whether the top-degree rational cohomology H^{vcd(SL_n(R))}(SL_n(R); Q) vanishes when R is a non-Euclidean PID number ring.

Background

Lee–Szczarba proved top-degree vanishing for SL_n(R) when R is a Euclidean number ring, while Church–Farb–Putman established non-vanishing over non-PID number rings. These results together leave the case of non-Euclidean PID number rings unresolved.

Assuming GRH, there are exactly four such imaginary quadratic rings of integers (O_{−19}, O_{−43}, O_{−67}, O_{−163}); subsequent work shows non-vanishing for SL_{2n}(O_d) for d ∈ {−43, −67, −163}, highlighting O_{−19} as a remaining case.

References

Hence, combining the vanishing result of Lee--Szczarba (\cref{eq_lee_szcarba}) with \cref{eq:non_vanishing_SLn}, the only number rings for which it is unknown whether \mathrm{SL}_n(\mathcal{O}) has vanishing top-degree cohomology are non-Euclidean PIDs.

— (Non-)Vanishing of high-dimensional group cohomology (2404.15026 - Brück, 23 Apr 2024) in Section 5 (SL_n over other number rings), paragraph “SL_n over non-Euclidean PIDs”