l 2 -Betti numbers for group theorists

Transcription

1 1/7 l 2 -Betti numbers for group theorists A minicourse in 3 parts 2nd lecture Roman Sauer Karlsruhe Institute of Technology Copenhagen, October 2013

2 The von Neumann dimension dim Γ Finite-dimensional vector spaces Let W C n be a subspace and pr W : C n C n be the projection onto W. Then dim C (W ) = tr Mn(C)(pr W ). 1/7

3 1/7 The von Neumann dimension dim Γ Finite-dimensional vector spaces Let W C n be a subspace and pr W : C n C n be the projection onto W. Then dim C (W ) = tr Mn(C)(pr W ). The von Neumann trace Let L(Γ) be the algebra of Γ-equivariant bounded operators on l 2 (Γ) (von Neumann algebra of Γ). For T L(Γ) define: tr Γ (T ) = Te, e l2 (Γ). It satisfies tr Γ (ST ) = tr Γ (TS) and extends to M n (L(Γ)). Hilbert Γ-modules A Hilbert Γ-module is a Hilbert space with an isometric linear Γ-action such that there is an Γ-equivariant isometric embedding H l 2 (Γ) n. dim Γ (H) := tr Mn(L(Γ))(pr H )

4 l 2 -Betti numbers Definition Let X be a cocompact free Γ-CW complex. Its l 2 -cohomology H(2) i (X ) is a Hilbert Γ-module via the embedding: We define the l 2 -Betti numbers as: H i (2) (X ) = ker( i ) l 2 (Γ) n. β (2) ( i (X ) = dim Γ H (2) i (X )) β (2) ( i (Γ) = dim Γ Hi (2) (EΓ) ) Homology versus cohomology Alternatively, we could define β (2) i by reduced homology. The Laplace operators are the same for homology and cohomology. Properties l 2 -Betti numbers satisfy equivariant homotopy invariance, a Künneth formula, and a Euler-Poincare formula... 2/7

5 3/7 A question by Atiyah M.P. ATIYAH These real Betti numbers appear to deserve further study. Some natural questions are : (i) Triangulate X and compute the simplicial cohomology (. cocycles/closure of cobounfor the lifted triangulation uslng of X? daries). Are these groups r-isomorphic to our t (ii) If the answer to (i) is yes, are the Bqr(X) homotopy invariants of X? (iii) A priori the num b ers Bq(X) r are real. Give examples where they are not integral and even perhaps irrational. REFERENCES

6 4/7 Conjectures Atiyah conjecture Let Γ be a torsionfree group. Then the l 2 -Betti numbers of a free cocompact Γ-CW complex are in N {0}. Algebraic Atiyah conjecture Let Γ be a torsionfree group. Let A M m,n (ZΓ). Then dim Γ ( ker ( l 2 (Γ) m A l 2 (Γ) n)) N {0}. Zero divisor conjecture Let Γ be a torsionfree group. The group ZΓ has no zero-divisors.

7 4/7 Conjectures Atiyah conjecture Let Γ be a torsionfree group. Then the l 2 -Betti numbers of a free cocompact Γ-CW complex are in N {0}. Algebraic Atiyah conjecture Let Γ be a torsionfree group. Let A M m,n (ZΓ). Then dim Γ ( ker ( l 2 (Γ) m A l 2 (Γ) n)) N {0}. Zero divisor conjecture Let Γ be a torsionfree group. The group ZΓ has no zero-divisors. Relations Atiyah algebraic Atiyah ZD conjecture

8 5/7 Free groups: Linnell s proof Fredholm module A Fredholm Γ-module consists of -homomorphisms ρ ± : L(Γ) B(H) such that ρ + (a) ρ (a) has finite rank for every a CΓ. Construct a Fredholm module with trace property We construct a specific Fredholm F 2 -module ρ ± such that tr F2 (a) = tr ( ρ + (a) ρ (a) ) =: τ(a) for all a L(F 2 ). }{{}}{{} F 2-trace usual trace

9 Free groups: Linnell s proof Fredholm module A Fredholm Γ-module consists of -homomorphisms ρ ± : L(Γ) B(H) such that ρ + (a) ρ (a) has finite rank for every a CΓ. Construct a Fredholm module with trace property We construct a specific Fredholm F 2 -module ρ ± such that tr F2 (a) = tr ( ρ + (a) ρ (a) ) =: τ(a) for all a L(F 2 ). Let (V, E) be the Cayley graph of F 2. Let f : V E {pt} { first edge on geodesic from x to x 0 if x x 0 f (x) = pt otherwise Standard reps ρ + : L(F 2 ) B(l 2 (V )) and ρ : L(F 2 ) B(l 2 (E) C) }{{} =l 2 (Γ) ρ := F 1 ρ F : L(F 2 ) B(l 2 (V )) with F induced by f. 5/7

10 6/7 Free groups: Linnell s proof Integrality lemma Let p, q B(H) be projections for which p q has finite rank. Then tr(p q) Z. Proof. p, q commute with (p q) 2, thus respect the eigenspace decomposition of (p q) 2 : H = E λ ker(p q) 2 }{{} λ 0 =ker(p q) This implies: tr(p q) = λ 0 tr( p Eλ ) λ 0 tr( q Eλ ) Z

11 6/7 Free groups: Linnell s proof Integrality lemma Let p, q B(H) be projections for which p q has finite rank. Then tr(p q) Z. Proof. p, q commute with (p q) 2, thus respect the eigenspace decomposition of (p q) 2 : H = E λ ker(p q) 2 }{{} λ 0 =ker(p q) This implies: tr(p q) = λ 0 tr( p Eλ ) λ 0 tr( q Eλ ) Z Finite rank lemma Let a, b B(H) such that a b has finite rank. Then pr ker(a) pr ker(b) has finite rank. Proof. It is easy to see that pr ker(a) and pr ker(b) agree on ker(a b). Since a b has finite rank, ker(a b) is finite-dimensional.

12 7/7 A few comments on the status of the conjectures Linnell: (Strong) Atiyah conjecture for a certain class of groups that contains free groups and elementary amenable groups with upper bounds on orders of finite subgroups. This builds on earlier work of Kropholler-Linnell-Moody who proved the ZD conjecture for torsionfree elementary amenable groups. A lot of positive results by Linnell, Schick and others... The Grigorchuk-Schick-Zuk counterexample to the strong Atiyah conjecture. Austin s counterexample to the rationality of L 2 -Betti numbers.