** 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

are isomorphisms.

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.

**Remarks**

- 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.

**Examples**.

- 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 3Let be the -skeleton of .

Theorem 4 (Guentner-Willett-Yu)The following are equivalent,

- There exists such that is almost equivariant.
- .

Example 1Hyperbolic 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 5Given 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.