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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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