Residual finiteness and cuspidal cohomology of Picard modular surfaces

Richard M. Hill

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arXiv · 2609.07639

Residual finiteness and cuspidal cohomology of Picard modular surfaces

Richard M. Hill

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Abstract

We prove that, for every non-uniform arithmetic lattice in $\mathrm{SU}(2,1)$, 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 $\Gamma$ for which $$H^1_{\mathrm{cusp}}(\Gamma\backslash\mathbb B^2,\mathbb C)\ne 0.$$ In particular, the first inner cohomology of this ball quotient is non-zero. The proof uses Rogawski's endoscopic classification for $\mathrm{U}(3)$. 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.

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Richard M. Hill. 2026-09-07. Residual finiteness and cuspidal cohomology of Picard modular surfaces. https://arxiv.org/abs/2609.07639

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