Let Λ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of C * (Λ) on certain separable Hilbert spaces of the form L 2 (X, µ), by introducing the notion of a Λ-semibranching function system (a generalization of the semibranching function systems studied by Marcolli and Paolucci). In particular, if Λ is aperiodic, we obtain a faithful representation ofwhere M is the Perron-Frobenius probability measure on the infinite path space Λ ∞ recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a Λ-semibranching function system gives rise to KMS states for C * (Λ). For the higher-rank graphs of Robertson and Steger, we also obtain a representation of, where X is a fractal subspace of [0, 1] by embedding Λ ∞ into [0, 1] as a fractal subset X of [0,1]. In this latter case we additionally show that there exists a KMS state for C * (Λ) whose inverse temperature is equal to the Hausdorff dimension of X. Finally, we construct a wavelet system for L 2 (Λ ∞ , M ) by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.2010 Mathematics Subject Classification: 46L05.

In this paper, we present a new way to associate a finitely summable spectral triple to a higher-rank graph Λ, via the infinite path space Λ ∞ of Λ. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of Λ ∞ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary -Bratteli diagrams, in order to associate a family of ultrametric Cantor sets, and their associated Pearson-Bellissard spectral triples, to a finite, strongly connected higher-rank graph Λ. We then study the zeta function, abscissa of convergence, and Dixmier trace associated to the Pearson-Bellissard spectral triples of these Cantor sets, and show these spectral triples are -regular in the sense of Pearson and Bellissard. We obtain an integral formula for the Dixmier trace given by integration against a measure , and show that is a rescaled version of the measure on Λ ∞ which was introduced by an Huef, Laca, Raeburn, and Sims. Finally, we investigate the eigenspaces of a family of Laplace-Beltrami operators associated to the Dirichlet forms of the spectral triples. We show that these eigenspaces refine the wavelet decomposition of 2 (Λ ∞ , ) which was constructed by Farsi et al. 2.5 below. The space of infinite paths  of a stationary -Bratteli diagram  is often a Cantor set, enabling us to study its associated Pearson-Bellissard spectral triple. Indeed, if the matrices 1 , … , are the adjacency matrices for a -graph Λ, then the space of infinite paths in Λ is homeomorphic to the Cantor set  (also called ). In other words, the Pearson-Bellissard spectral triples for stationary -Bratteli diagrams can also be viewed as spectral triples for higher-rank graphs.We then proceed to study, in Section 3, the geometrical information encoded by these spectral triples. Theorem 3.14 establishes that the Pearson-Bellissard spectral triple associated to (  Λ , ) is finitely summable, with dimension ∈ (0, 1). Section 3.3 focuses on the Dixmier traces of the spectral triples, and establishes both an integral formula for the Dixmier trace (Theorems 3.23 and 3.28) and a concrete expression for the measure induced by the Dixmier trace (Theorem 3.26). These computations also reveal that the ultrametric Cantor sets (  Λ , ) are -regular in the sense of [59, Definition 11]. Other settings in the literature in which spectral triples on Cantor sets admit an integral formula for the Dixmier trace include [13,47,17,14].In full generality, Dixmier traces are defined on the Dixmier-Macaev (also called Lorentz) ideal  1,∞ ⊆ () inside the compact operators and are computed using a generalized limit (roughly speaking, a linear functional that lies between lim sup and lim inf). Although the theory of Dixmier traces can be quite intricate, many of the computations simplify substantially in our setting, and so our treatment of the general theory will be brief; we refer the interested reader to the extensive literature on Dixmier traces and other singular traces (cf. [19, 55, 54, 1...

We introduce the -Euler-Satake characteristics of a general orbifold Q presented by an orbifold groupoid G , extending to orbifolds that are not global quotients the generalized orbifold Euler characteristics of Bryan-Fulman and Tamanoi. Each of these Euler characteristics is defined as the Euler-Satake characteristic of the space of -sectors of the orbifold where is a finitely generated discrete group. We study the behavior of these Euler characteristics under product operations applied to the group as well as the orbifold and establish their relationships to existing Euler characteristics for orbifolds. As applications, we generalize formulas of Tamanoi, Wang and Zhou for the Euler characteristics and Hodge numbers of wreath symmetric products of global quotient orbifolds to the case of quotients by compact, connected Lie groups acting locally freely, in particular including all closed, effective orbifolds. 22A22, 55S15; 58E40, 55N91

Abstract. We introduce the notion of regular symplectomorphism and graded regular symplectomorphism between singular phase spaces. Our main concern is to exhibit examples of unitary torus representations whose symplectic quotients cannot be graded regularly symplectomorphic to the quotient of a symplectic representation of a finite group, while the corresponding GIT quotients are smooth. Additionally, we relate the question of simplicialness of a torus representation to Gaussian elimination.

Abstract. We introduce a complete obstruction to the existence of nonvanishing vector fields on a closed orbifold Q. Motivated by the inertia orbifold, the space of multi-sectors, and the generalized orbifold Euler characteristics, we construct for each finitely generated group Γ an orbifold called the space of Γ-sectors of Q. The obstruction occurs as the Euler-Satake characteristics of the Γ-sectors for an appropriate choice of Γ; in the case that Q is oriented, this obstruction is expressed as a cohomology class, the Γ-Euler-Satake class. We also acquire a complete obstruction in the case that Q is compact with boundary and in the case that Q is an open suborbifold of a closed orbifold.

In this article we provide an identification between the wavelet decompositions of certain fractal representations of C * algebras of directed graphs of M. Marcolli and A. Paolucci [19], and the eigenspaces of Laplacians associated to spectral triples constructed from Cantor fractal sets that are the infinite path spaces of Bratteli diagrams associated to the representations, with a particular emphasis on wavelets for representations of Cuntz C * -algebras O D . In particular, in this setting we use results of J. Pearson and J. Bellissard [20], and A. Julien and J. Savinien [15], to construct first the spectral triple and then the Laplace-Beltrami operator on the associated Cantor set. We then prove that in certain cases, the orthogonal wavelet decomposition and the decomposition via orthogonal eigenspaces match up precisely. We give several explicit examples, including an example related to a Sierpinski fractal, and compute in detail all the eigenvalues and corresponding eigenspaces of the Laplace-Beltrami operators for the equal weight case for representations of O D , and in the uneven weight case for certain representations of O 2 , and show how the eigenspaces and wavelet subspaces at different levels first constructed in [8] are related.2010 Mathematics Subject Classification: 46L05.

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