Trace theorem for quasi-Fuchsian groups

Abstract

Abstract. We complete the proof of the Trace Theorem in the quantized calculus for quasi-Fuchsian group which was stated and sketched, but not fully proved, on pp. 322-325 in the book "Noncommutative Geometry"of the first author.

Quantum Differentiability on Noncommutative Euclidean Spaces

“…Connes introduced the quantised calculus in [8] as a replacement for the algebra of differential forms for applications in a noncommutative setting, and afterwards this point of view found application to mathematical physics [9]. Connes successfully applied his quantised calculus in providing a formula for the Hausdorff measure of Julia sets and for limit sets of Quasi-Fuchsian groups in the plane [10, Chapter 4, Section 3.γ] (for a more recent exposition see [17,14]).…”

The Connes character formula for locally compact spectral triples

“…The main technical innovation of this paper concerns a certain integral representation for the difference of complex powers of positive operators, which originally appeared in [25] and which is reproduced here as Theorem 5.2.1.…”

Conformal trace theorem for Julia sets of quadratic polynomials

“…By applying the method of complex interpolation as in [5,Lemma 3.4], it suffices to prove (A.1) for p = p 0 and p = 2. Firstly, for p = 2, we note that the spaces A 1/2 2 and C 1/2 2 are Hilbert spaces, and that the functions e n (z) = z n , n ≥ 0, are orthogonal in both spaces with dense linear span.…”

“…Both the absolute value and the raising to real powers do make sense in the quantized calculus, where moreover the role of the derivative of Z is played by the quantized differential [F, Z ], where Z is viewed as the multiplication operator M Z and F is the Hilbert transform. Thus, |d Z | p makes sense as an operator while as stated above the role of the integration is played by the Theorem 1.1 should be compared with [5,Theorem 1.1], which concerns geometric measures on limit sets of finitely generated quasi-Fuchsian groups. The statement of the result is very similar; however, it should be noted that the methods of proof used in this text are completely different to those used in [5].…”

“…Thus, |d Z | p makes sense as an operator while as stated above the role of the integration is played by the Theorem 1.1 should be compared with [5,Theorem 1.1], which concerns geometric measures on limit sets of finitely generated quasi-Fuchsian groups. The statement of the result is very similar; however, it should be noted that the methods of proof used in this text are completely different to those used in [5]. We follow a proof outlined by the first named author in [2,Ch.…”