Geometric measure on quasi-fuchsian groups via singular traces. Part I.

Transcription

1 Geometric measure on quasi-fuchsian groups via singular traces. Part I. Dmitriy Zanin (in collaboration with A. Connes and F. Sukochev) University of New South Wales September 19, 2018 Dmitriy Zanin Geometric measure via singular traces September 19, / 22

2 Introduction Kleinian groups: basics We let SL(2, C) be the group of all 2 2 complex matrices with determinant 1. We identify the group PSL(2, C) = SL(2, C)/{±1} and its action on the complex sphere C by fractional linear transformations. The matrix ( ) g11 g g = 12 is identified with the mapping z g 11z + g 12. g 21 g 22 g 21 z + g 22 We use the following definition of a Kleinian group. Definition Let G PSL(2, C) be a discrete subgroup. We say that (a) (b) G is regular at the point z C if there exists a neighborhood U z such that g(u) U = for all but finitely many g G. G is Kleinian if it is regular at some point z C. Dmitriy Zanin Geometric measure via singular traces September 19, / 22

3 Introduction Limit set of a Kleinian group Definition Let G be a Kleinian group. (a) the set of all points at which G is regular is called a regular set of G. (b) the complement of the regular set is called a limit set of G. The limit set is denoted by Λ(G). In the theory of Kleinian groups it plays the same role as the Julia set in conformal dynamics. By definition, Λ(G) is a closed G invariant set. One can show that Λ(G) is nowhere dense. Usually, Λ(G) is a perfect set (i.e., every point is an accumulation point). Exceptions are so-called elementary Kleinian groups, whose limit sets contain at most 2 points. Dmitriy Zanin Geometric measure via singular traces September 19, / 22

4 Introduction Fuchsian and quasi-fuchsian groups Definition A Kleinian group G is called (a) (b) Fuchsian (of the first kind) if its limit set is a circle. quasi-fuchsian if its limit set is a Jordan curve. In this talk, we are mostly interested in finitely generated quasi-fuchsian groups. By analogue with the Julia sets, one would expect that a limit set of a quasi-fuchsian group is a fractal curve. In particular, one would expect that its Hausdorff dimension is strictly greater than 1. Dmitriy Zanin Geometric measure via singular traces September 19, / 22

5 Introduction Hausdorff dimension of the limit set Definition We say that the Hausdorff dimension of a set X C is less than q if, for every ɛ > 0, there exist balls (B(a i, r i )) i I such that X i I B(a i, r i ), i I r q i < ɛ. The infimum of all such q is called the Hausdorff dimension of a set X. Theorem Let G be a finitely generated quasi-fuchsian (but not Fuchsian) group. Its limit set Λ(G) has Hausdorff dimension strictly greater than 1. Dmitriy Zanin Geometric measure via singular traces September 19, / 22

6 Introduction Geometric measure Definition Let G be a Kleinian group. The measure ν on C is called p dimensional geometric (relative to G) if for every g G. d(ν g)(z) = g (z) p dν(z) We are particularly interested in the case when p is the Hausdorff dimension of Λ(G). Theorem (Sullivan) Let G be a geometrically finite Kleinian group. If p is the Hausdorff dimension of Λ(G), then p < 2 and there exists a unique p dimensional geometric measure supported on Λ(G). Dmitriy Zanin Geometric measure via singular traces September 19, / 22

7 Introduction How to express geometric measure? Geometric measures are contructed by means of a complicated process. It is strongly desirable to have a simple and accessible construction. Question Is there any construction of geometric measure from the first principles? In this talk, we propose such a construction for the special case of finitely generated quasi-fuchsian groups. The only components of these construction are 1 Riemann mapping theorem 2 Hilbert transform 3 trace on L 1, Dmitriy Zanin Geometric measure via singular traces September 19, / 22

8 Preliminary material: traces General information Let L be the algebra of all bounded operators on a given (separable, infinite dimensional) Hilbert space H. An operator is called compact if it can be approximated (in norm topology, with any given precision) by a finite rank operator. Spectrum of a compact operator consists of non-zero eigenvalues of finite multiplicity converging to 0 (which may or may not be an eigenvalue). If A is compact, then A is compact. Eigenvalues of A are called singular values of A. For a compact operator A, we define its singular value sequence µ(a) = (µ(k, A)) k 0 by arranging the eigenvalues of A in the decreasing order and taking them with multiplicities. Dmitriy Zanin Geometric measure via singular traces September 19, / 22

9 Preliminary material: traces Ideals and infinitesimals An ideal I in L is a linear subspace (usually not closed in norm) such that A I and B L implies that AB, BA I. Ideal is called principal if it is generated by a single element. Every non-trivial ideal in L consists of compact operators. 1 Principal ideal generated by the diagonal operator diag(( ) (k+1) 1/p k 0 ) is called L p,. For every p > 0, it is quasi-banach (see next page). Equivalently, { } L p, = A L : µ(k, A) = O((k + 1) 1 p ). In Connes ideology, these are infinitesimals of order 1 p. Dmitriy Zanin Geometric measure via singular traces September 19, / 22

10 Preliminary material: traces Quasi-Banach ideals Definition An ideal I in L is called quasi-banach when equipped with a complete quasi-norm I such that AB I, BA I A I B. For example, a natural quasi-norm on the ideal L p, is given by the formula A p, = sup(k + 1) 1 p µ(k, A). k 0 When equipped with this quasi-norm, L p, becomes a quasi-banach ideal. In fact, for p > 1 its natural quasi-norm is equivalent to a norm. We let (L p, ) 0 to be the closure of finite rank operators with respect to the quasi-norm p,. Dmitriy Zanin Geometric measure via singular traces September 19, / 22

11 Preliminary material: traces Traces on ideals Definition Let I be an ideal in L. Linear functional ϕ : I C is called trace if ϕ(ab) = ϕ(ba), A I, B L. Equivalently, for all unitary U L, ϕ(u 1 AU) = ϕ(a), A I. Dmitriy Zanin Geometric measure via singular traces September 19, / 22

12 Preliminary material: traces Traces on ideals: L p,, p > 1 Example For p > 1, ideal L p, does not carry any trace. Proof. If X L p,, then there exist (X k ) 20 k=1 L p, and (Y k ) 20 k=1 L such that 20 X = [X k, Y k ]. k=1 Hence, for every trace ϕ, we have 20 ϕ(x ) = ϕ(x k Y k ) ϕ(y k X k ) = 0. k=1 Dmitriy Zanin Geometric measure via singular traces September 19, / 22

13 Preliminary material: traces Traces on ideals: L 1, Ideal L 1, carries a plethora of traces. The most famous one is due to Dixmier. Definition Let ω be a free ultrafilter on Z +. The mapping Tr ω : A lim n ω 1 log(n + 2) n µ(k, A), 0 A L 1, k=0 is additive. Its linear extension to L 1, is called Dixmier trace. Dmitriy Zanin Geometric measure via singular traces September 19, / 22

14 Preliminary material: traces Traces on L 1, : further properties 1 Every Dixmier trace is positive. 2 Every positive trace on L 1, is continuous. 3 Every continuous trace on L 1, is a linear combination of positive ones. 4 There are continuous traces on L 1, which are not Dixmier traces. 5 There are traces on L 1, which fail to be continuous. 6 There are 2 2N continuous traces on L 1,. 7 Every trace on L 1, vanishes on L 1. 8 Every continuous trace on L 1, vanishes on (L 1, ) 0. Dmitriy Zanin Geometric measure via singular traces September 19, / 22

15 Statement of the main result Morphism from C(Λ(G)) to C(T) For quasi-fuchsian group G, the limit set is a Jordan curve. Hence, it divides the complex sphere C into 2 simply connected parts: Λ(G) int and Λ(G) ext. Riemann mapping theorem says that a simply connected domain Λ(G) int is conformally equivalent to the unit disk D. Let Z : D Λ(G) int be the conformal equivalence. By Caratheodory theorem, Z extends to a homeomorphism Z : T Λ(G). We now have a natural morphism C(Λ(G)) C(T) f f Z. Dmitriy Zanin Geometric measure via singular traces September 19, / 22

16 Statement of the main result The Hilbert transform The Hilbert space L 2 (T) is defined with respect to the arc-length measure (the Haar measure). There is the trigonometric orthonormal basis for L 2 (T), e n (z) = z n, n Z, z T. The Hilbert transform F is defined on the basis e n by Fe n = sgn(n)e n. Dmitriy Zanin Geometric measure via singular traces September 19, / 22

17 Statement of the main result Main theorem The assertion below was proposed (without rigorous proof) in the Noncommutative Geometry by Connes. Complete proof appeared in a recent paper by the authors. Theorem Let G be a finitely generated quasi-fuchsian group without parabolic elements ane let p be the Hausdorff dimension of Λ(G). (a) (b) [F, M Z ] L p, for every continuous trace ϕ on L 1, and for every f C(Λ(G)) we have ϕ(m f Z [F, M Z ] p ) = c ϕ f (z)dν(z), Λ(G) where ν is the unique p dimensional geometric measure on Λ(G). (c) there is a positive trace ϕ on L 1, such that c ϕ > 0. Dmitriy Zanin Geometric measure via singular traces September 19, / 22

18 Quantised calculus Where [F, M h ] belongs? It is a classical result by Kronecker that [F, M h ] is finite rank if and only if h is a rational function. Nehari proved that [F, M h ] is bounded if and only if h has bounded mean oscillation (in short, h BMO). It is immediate that [F, M h ] L 2 if and only if h belongs to the Sobolev space W 1 2,2. Question When [F, M h ] belongs to L p,? Dmitriy Zanin Geometric measure via singular traces September 19, / 22

19 Quantised calculus Peller theorem The answer to the question above can be derived from the results by Peller. Theorem The operator [F, M h ] belongs to L p if and only if h belongs to a Besov space B 1 p p. We want to apply this result to the function h = Z. For this function, we only know the behavior inside D, while its behavior on the boundary is much harder to investigate. It, therefore, makes sense to restate Peller s result in terms of analytic extension of the function h to the unit disk. Dmitriy Zanin Geometric measure via singular traces September 19, / 22

20 Quantised calculus Restatement of Peller s result for L p, 1 p 2 Theorem Suppose h admits an extension to the unit disk. The operator [F, h] belongs to L 1 if and only if h L 1 (D). It is immediate that [F, M h ] L 2 if and only if h L 2 (D, (1 z 2 )dzd z). An intepolation argument yields Theorem Suppose h admits an extension to the unit disk. The operator [F, h] belongs to L p if and only if h L p (D, (1 z 2 ) 2p 2 dzd z). Dmitriy Zanin Geometric measure via singular traces September 19, / 22

21 Quantised calculus Peller-type result for L p,, 1 < p < 2 Preceding theorem can be simplified for p > 1 as follows: Theorem Suppose h admits an extension to the unit disk. The operator [F, h] belongs to L p if and only if h L p (D, (1 z 2 ) p 2 dzd z). An interpolation argument yields Theorem Suppose h admits an extension to the unit disk. The operator [F, h] belongs to L p, if and only if k L p, (D, (1 z 2 ) 2 dzd z). Here, k(z) = h (z)(1 z 2 ), z < 1. Dmitriy Zanin Geometric measure via singular traces September 19, / 22

22 Quantised calculus TO BE CONTINUED Dmitriy Zanin Geometric measure via singular traces September 19, / 22