PreprintarXiv licencearXiv
Residual finiteness and cuspidal cohomology of Picard modular surfaces
Abstract
We prove that, for every non-uniform arithmetic lattice in SU(2,1), its inverse images in the universal cover and in all connected finite covers are residually finite. The key new input is that every commensurability class of such lattices contains a congruence arithmetic lattice Γ for which H1cusp(Γbackslash𝔹2,ℂ)≠ 0. In particular, the first inner cohomology of this ball quotient is non-zero. The proof uses Rogawski's endoscopic classification for U(3). A cohomological criterion proved previously by the author then gives the residual-finiteness result. Residual finiteness also yields multiplier systems of arbitrary denominator on suitable finite-index subgroups.
Related
- On the cuspidal cohomology of Iwahori congruence subgroups of SL(3, ℤ)
- Tate cohomology of unipotent cuspidal representations of finite groups of Lie type
- Cuspidal Class Number Formulae for Intermediate Modular Curves of Prime Level
- The Serre-Grothendieck Finiteness Theorem for the cohomology of coherent sheaves
- On the residual finiteness of the non-abelian exterior square of wreath products and the Grigorchuk group
BibTeX
@misc{hill2026residual,
title = {{Residual finiteness and cuspidal cohomology of Picard modular surfaces}},
author = {Richard M. Hill},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07639},
eprint = {2609.07639},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07639},
}
RIS
TY - UNPB
TI - Residual finiteness and cuspidal cohomology of Picard modular surfaces
AU - Richard M. Hill
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07639
UR - https://arxiv.org/abs/2609.07639
AB - We prove that, for every non-uniform arithmetic lattice in SU(2,1), its inverse images in the universal cover and in all connected finite covers are residually finite. The key new input is that every commensurability class of such lattices contains a congruence arithmetic lattice Γ for which H^1_cusp(Γbackslash𝔹^2,ℂ)≠ 0. In particular, the first inner cohomology of this ball quotient is non-zero. The proof uses Rogawski's endoscopic classification for U(3). A cohomological criterion proved previously by the author then gives the residual-finiteness result. Residual finiteness also yields multiplier systems of arbitrary denominator on suitable finite-index subgroups.
ER -