ON INTEGRAL ASSEMBLY MAPS FOR LATTICES IN SL3

This paper proves the integral Novikov conjecture in algebraic K-theory for lattices in the special linear group SL3, a semisimple Lie group of rank 2. The group SL3 has been used extensively as a trial range in extending analysis on locally symmetric spaces to “higher Q-ranks”. In a similar way, our argument uses a refinement of the methods previously successful where geometry of the group possessed some manifestation of nonpositive curvature [3, 4, 8, 9]. Theorem. If Γ is a torsion-free lattice in SL3 and R is an arbitrary ring, the in-tegral assembly map α: h(Γ, K(R)) → K(R[Γ]) from the homology of the group Γ with coefficients in the K-theory spectrum K(R) to the K-theory of the group ring R[Γ] is a split injection. Here K(A) stands for the nonconnective K-theory spectrum of the ring A. A major geometric component of the proof is the construction of a new Γ-equivariant compactification of the associated symmetric space which also contains the Borel-Serre enlargement of the symmetric space and the study of its properties