## Notes of Arthur Bartels’ first Southampton lecture 27-03-2017

The Farrell-Jones Conjecture and (coarse) flow spaces, I

Today, actions on ERs. Tomorrow, flows.

Theorem 1 (Farrell-Jones) ${M}$ closed manifold of non-positive curvature, dimension at least 5. Then ${M}$ is topologically rigid: anay manifold homotopy equivalent to ${M}$ is homeomorphic to ${M}$ (i.e. Borel’s conjecture holds for such manifolds).

1. It uses surgery theory heavily, especially calculations in algebraic K and L-theory.
2. It use the geodesic flow on ${\tilde M}$.
3. Cyclic subgroups of ${\pi_1(M)}$ play an important role, as stabilizers of stable leaves of the flow.

This result has led to the

Farrell-Jones Conjecture. Let ${R}$ be a ring, ${G}$ a group. Denote by ${VCyc}$ the collection of virtually cyclic subgroups of ${G}$. Then the assembly maps

$\displaystyle \begin{array}{rcl} H^*_G(E_{VCyc}G;\mathbb{K}_R)\rightarrow K_*(R[G]), \quad H^*_G(E_{VCyc}G;\mathbb{L}_R)\rightarrow L_*(R[G]), \end{array}$

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

1. Suppose the FJC holds for ${G}$, and that all ${K_* R[V]}$ vanish. Then ${K_*(R[G])}$ vanishes.
2. Let ${G}$ be a torsion free group, ${M}$ a manifold of dimension at least 5 whose fundamental group is equal to ${G}$. The FJC tells us that the set of homotopy classes of homotopy equivalences of closed manifolds ${N}$ to ${M}$ is

$\displaystyle \begin{array}{rcl} H_{n+1}(BG,M,\mathbb{L}_{{\mathbb Z}}). \end{array}$

3. There is a version of FJC relative to a family ${\mathcal{F}}$ of subgroups.

0.1. Euclidean retracts

An ER is a compact space that embeds as a retract in ${{\mathbb R}^n}$. There is an abstract characterization: finite dimensional, contractible, compact ANR’s.

Brouwer: any action of ${{\mathbb Z}}$ on an ER has a fixed point.

What about ${{\mathbb Z}^2}$? It turns out that ${{\mathbb Z}^2}$ admits a simplicial action on an ER without global fixed-point.

Examples.

1. If ${M}$ has non-positive curvature, Its fundamental group ${G}$ acts on the visual compactification ${\tilde M \cup S_\infty\simeq D^n}$. If furthermore ${K<0}$, isotropy is cyclic.
2. ${Let, G}$ be hyperbolic. Then ${G}$ 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.
3. Let ${G}$ 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.

4. Let ${G=Out(F_n)}$. Bestvina-Horbez construct an equivariant compactification of Outer space ${CV_n}$, where isotropy are virtually cyclic of smaller ${Out(F_m)}$.

1. Amenable actions

We compare ${X}$ with the more handy space of measures on ${X}$.

An action of A group ${G}$ on a space ${X}$ is amenable if there exists a sequence of maps ${f_n:X\rightarrow \Delta(G)}$ (the full simplex of probability measures on ${G}$ with finite support), which is almost equivariant (in ${\ell_1(g))}$.

Theorem 2 (Higson-Roe, Yu) If the action of ${G}$ on its Stone-\v Cech compactfication ${\beta G}$ is amenanble, then the analytic Novikov conjecture for ${G}$.

Example. Let ${F_n}$ act on its boundary. Let ${e-g_1-g_2-\cdots}$ be a geodesic ray. Set

$\displaystyle \begin{array}{rcl} f_n(\xi):=\sum_{i=0}^n \frac{1}{n+1}\delta_{g_i}. \end{array}$

are almost equivariant due to 0-hyperbolicity.

Definition 3 Let ${\Delta^N(G)}$ be the ${N}$-skeleton of ${\Delta(G)}$.

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

1. There exists ${N}$ such that ${f_n:\beta G\rightarrow \Delta^N(G)}$ is almost equivariant.
2. ${asdim(G)<\infty}$.

Example 1 Hyperbolic groups, Mapping class groups (Bestvina-Bromberg-Fujiwara) have finite asymptotic dimension. Question is open for ${Out(F_n)}$.

This cannot happen if the ${G}$ action on ${X}$ has infinite isotropy.

This is why we introduce a relative version.

Definition 5 Given a family ${\mathcal{F}}$ of subgroups, let ${S}$ be the disjoint union of quotients ${G/F}$, ${F\in \mathcal{F}}$, ${E_{\mathcal{F}}G}$ the joint of infinitely many copies of ${S}$, ${E_{\mathcal{F}}^N G}$ its ${N}$-skeleton. Say that the ${G}$-action on ${X}$ is ${N-\mathcal{F}}$-amenable if there exists an almost equivariant sequence of maps ${f_nX\rightarrow E_{\mathcal{F}}^N(G)}$.

Theorem 6 (Bartels-Luck-Reich) If ${G}$ admits an ${N-\mathcal{F}}$-amenable action on some ER, the ${G}$ action satfisfies the FJC relative to ${\mathcal{F}}$.

Examples where our theorem applies:

1. ${G}$ hyperbolic, ${\mathcal{F}=VCyc}$.
2. ${G}$ relatively hyperbolic, ${\mathcal{F}=VCyc}$ and parabolics.
3. ${G=MCG}$, ${\mathcal{F}=VCyc}$ and smaller MCG’s (Bestvina). With induction, this proves FJC fro MCG.

We do not know wether FJC pass to finite index subgroups.