The homological torsion of PSL_2 of the imaginary quadratic integers

Denote by Q(sqrt{-m}), with m a square-free positive integer, an imaginary quadratic number field, and by A its ring of integers. The Bianchi groups are the groups SL_2(A). We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We expose a novel technique, the torsion subcomplex reduction, to obtain these invariants. We use it to explicitly compute the integral group homology of the Bianchi groups. Furthermore, this correspondence facilitates the computation of the equivariant K-homology of the Bianchi groups. By the Baum/Connes conjecture, which is verified by the Bianchi groups, we obtain the K-theory of their reduced C*-algebras in terms of isomorphic images of their equivariant K-homology.

💡 Research Summary

The paper investigates the integral group homology of the Bianchi groups PSL₂(𝒪), where 𝒪 is the ring of integers in an imaginary quadratic field ℚ(√−m) with square‑free m. By exploiting the natural action of these groups on hyperbolic three‑space ℍ³, the author constructs a refined cellular complex in which each cell is fixed pointwise by its stabilizer. The central methodological innovation is the “torsion subcomplex reduction”: for each prime ℓ = 2, 3 (the only orders of non‑trivial finite elements in PSL₂(𝒪)), one extracts the subcomplex consisting of cells whose stabilizers contain ℓ‑torsion, then collapses configurations where a vertex is incident to exactly two ℓ‑torsion edges into a single edge. Lemmas and theorems (notably Lemma 16, Theorem 3, and Theorem 2) establish that the ℓ‑primary part of the homology in degrees greater than the virtual cohomological dimension (which is 2) depends solely on the homeomorphism type of this reduced ℓ‑torsion subcomplex.

Using the refined complex, the author applies an equivariant Leray–Serre spectral sequence whose E₂‑page is built from the homology of the finite stabilizer groups (which are cyclic of order 1, 2, 3, the Klein four group, S₃, or A₄). The differentials on the first page are completely determined by the incidence relations among cells and the inclusion maps of stabilizers. The spectral sequence collapses at a low page, yielding explicit descriptions of the integral homology: for the four non‑Euclidean principal ideal domain cases m ∈ {19, 43, 67, 163}, one has H₁ ≅ ℤ^{β₁} with β₁ = 1, 2, 3, 7 respectively, H₂ ≅ ℤ^{β₁−1} ⊕ ℤ/4 ⊕ ℤ/2 ⊕ ℤ/3, and in all higher degrees the homology is a direct sum of copies of ℤ/2 and ℤ/3. The numbers of copies are encoded in the Poincaré series \