Geometry and Topology Seminar

Let K be a number field with class number 1. We prove that if K admits a real embedding K ↪ R, then H^vcd(SL_nO_K; Q) = 0, where vcd is the virtual cohomological dimension of the lattice SL_nO_K. To do this, we prove that the homology of the Tits building for SL_nK is generated by O_K-integral apartments; in fact, this holds for a wide class of Dedekind domains of arithmetic type, including Z[i, 1/3] ⊂ Q(i) and F_q[T^±1] ⊂ F_q(T). The key technical ingredient is that the complex of partial bases for (O_K)ⁿ is Cohen–Macaulay; this complex has previously been used to prove that the homology of such lattices stabilizes as n → ∞. In contrast, we prove that the codimension-0 Betti number of SL_nO_K grows exponentially when the class number of K is greater than 2. Joint work with Benson Farb and Andrew Putman.