The Farrell-Jones Conjecture and (coarse) flow spaces, I
Today, actions on ERs. Tomorrow, flows.
Theorem 1 (Farrell-Jones) closed manifold of non-positive curvature, dimension at least 5. Then is topologically rigid: anay manifold homotopy equivalent to is homeomorphic to (i.e. Borel’s conjecture holds for such manifolds).
- It uses surgery theory heavily, especially calculations in algebraic K and L-theory.
- It use the geodesic flow on .
- Cyclic subgroups of play an important role, as stabilizers of stable leaves of the flow.
This result has led to the
Farrell-Jones Conjecture. Let be a ring, a group. Denote by the collection of virtually cyclic subgroups of . Then the assembly maps
In other words, the K-theory of the group ring can be computed in terms of the K-theories of group rings of virtually cyclic subgroups.
- Suppose the FJC holds for , and that all vanish. Then vanishes.
- Let be a torsion free group, a manifold of dimension at least 5 whose fundamental group is equal to . The FJC tells us that the set of homotopy classes of homotopy equivalences of closed manifolds to is
- There is a version of FJC relative to a family of subgroups.
0.1. Euclidean retracts
An ER is a compact space that embeds as a retract in . There is an abstract characterization: finite dimensional, contractible, compact ANR’s.
Brouwer: any action of on an ER has a fixed point.
What about ? It turns out that admits a simplicial action on an ER without global fixed-point.
- If has non-positive curvature, Its fundamental group acts on the visual compactification . If furthermore , isotropy is cyclic.
- be hyperbolic. Then acts on its Rips complex, whose compactification is an ER (Betsvina-Mess), isotropy is virtually cyclic. For more general relatively hyperbolic groups, isotropy is virtually cyclic or a peripheral subgroup.
- Let be the mapping class group. It acts on Teichmuller space, which is homeomorphic to a closed ball, and its Thurston closure. Isotropy is virtually cyclic or a smaller mapping class group.
- Let . Bestvina-Horbez construct an equivariant compactification of Outer space , where isotropy are virtually cyclic of smaller .
1. Amenable actions
We compare with the more handy space of measures on .
An action of A group on a space is amenable if there exists a sequence of maps (the full simplex of probability measures on with finite support), which is almost equivariant (in .
Theorem 2 (Higson-Roe, Yu) If the action of on its Stone-\v Cech compactfication is amenanble, then the analytic Novikov conjecture for .
Example. Let act on its boundary. Let be a geodesic ray. Set
are almost equivariant due to 0-hyperbolicity.
Definition 3 Let be the -skeleton of .
Theorem 4 (Guentner-Willett-Yu) The following are equivalent,
- There exists such that is almost equivariant.
Example 1 Hyperbolic groups, Mapping class groups (Bestvina-Bromberg-Fujiwara) have finite asymptotic dimension. Question is open for .
This cannot happen if the action on has infinite isotropy.
This is why we introduce a relative version.
Definition 5 Given a family of subgroups, let be the disjoint union of quotients , , the joint of infinitely many copies of , its -skeleton. Say that the -action on is -amenable if there exists an almost equivariant sequence of maps .
Theorem 6 (Bartels-Luck-Reich) If admits an -amenable action on some ER, the action satfisfies the FJC relative to .
Examples where our theorem applies:
- hyperbolic, .
- relatively hyperbolic, and parabolics.
- , and smaller MCG’s (Bestvina). With induction, this proves FJC fro MCG.
We do not know wether FJC pass to finite index subgroups.