Farrell–Jones conjecture

The Farrell–Jones conjecture says that assembly maps from a group’s virtually cyclic subgroups recover its algebraic K-theory and L-theory, with coefficients in additive categories.

Connect

Assembly map: A map that combines local or subgroup-level data into a global homology or algebraic invariant. The conjecture asserts that these maps become isomorphisms for its specified family of subgroups.
Equivariant homology theory: A homology theory that accounts for a group action on spaces and the stabilizers of points. The assembly map is an equivariant homology map built from a group action.
F. Thomas Farrell: An American mathematician known for work in geometric topology and algebraic K-theory. Farrell introduced the conjectural program with Lowell Jones.
Whitehead group: An algebraic K-theory group that measures certain failures of homotopy equivalences to be simple. Assembly isomorphisms imply vanishing results for Whitehead groups in broad settings.
Hyperbolic group: A finitely generated group whose Cayley graph is hyperbolic in the sense of Gromov. The conjecture is known for broad classes of hyperbolic groups, making them key test cases.
Classifying space for a family of subgroups: A group space characterized by contractible fixed-point sets for subgroups in a chosen family and empty sets for others. Its virtually cyclic model supplies the domain of the conjectural assembly map.
Group ring: A ring whose elements are finite formal sums of group elements with coefficients in a ring. Its algebraic K- and L-theory forms the conjecture’s principal targets.
Lowell E. Jones: An American mathematician whose work includes geometric topology and the Farrell–Jones conjecture. Jones co-formulated the conjecture and developed its links to manifold topology.
Bass conjecture: A conjecture restricting the Hattori–Stallings traces of projective modules over group rings. Farrell–Jones results imply versions of this prediction about group-ring projectives.
CAT(0) group: A group acting properly and cocompactly by isometries on a CAT(0) space. Many CAT(0) groups satisfy the conjecture, while the full class remains a natural frontier.