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Register a new userCompressible Spaces and $mathcal{E}mathcal{Z}$-Structures
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Bestvina introduced a $mathcal{Z}$-structure for a group $G$ to generalize the boundary of a CAT(0) or hyperbolic group. A refinement of this notion, introduced by Farrell and Lafont, includes a $G$-equivariance requirement, and is known as an $mathcal{E}mathcal{Z}$-structure. In this paper, we show that fundamental groups of graphs of nonpositively curved Riemannian $n$-manifolds admit $mathcal{Z}$-structures and graphs of negatively curved or flat $n$-manifolds admit $mathcal{E}mathcal{Z}$-structures. This generalizes a recent result of the first two authors with Tirel, which put $mathcal{E}mathcal{Z}$-structures on Baumslag-Solitar groups and $mathcal{Z}$-structures on generalized Baumslag-Solitar groups.
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We generalize Bestvinas notion of a $mathcal{Z}$-boundary for a group to that of a coarse $mathcal{Z}$-boundary. We show that established theorems about $mathcal{Z}$-boundaries carry over nicely to the more general theory, and that some wished-for properties of $mathcal{Z}$-boundaries become theorems when applied to coarse $mathcal{Z}$-boundaries. Most notably, the property of admitting a coarse $mathcal{Z}$-boundary is a pure quasi-isometry invariant. In the process, we streamline both new and existing definitions by introducing the notion of a model $mathcal{Z}$-geometry. In accordance with the existing theory, we also develop an equivariant version of the above -- that of a coarse $Emathcal{Z}$-boundary.
For a small category $mathcal{D}$ we define fibrations of simplicial presheaves on the category $mathcal{D}timesDelta$, which we call localized $mathcal{D}$-left fibration. We show these fibrations can be seen as fibrant objects in a model structure, the localized $mathcal{D}$-covariant model structure, that is Quillen equivalent to a category of functors valued in simplicial presheaves on $mathcal{D}$, where the Quillen equivalence is given via a generalization of the Grothendieck construction. We use our understanding of this construction to give a detailed characterization of fibrations and weak equivalences in this model structure and in particular obtain a Yoneda lemma. We apply this general framework to study Cartesian fibrations of $(infty,n)$-categories, for models of $(infty,n)$-categories that arise via simplicial presheaves, such as $n$-fold complete Segal spaces. This, in particular, results in the Yoneda lemma and Grothendieck construction for Cartesian fibrations of $(infty,n)$-categories.
We report the spectral features of a phase-shifted parity and time ($mathcal{PT}$)-symmetric fiber Bragg grating (PPTFBG) and demonstrate its functionality as a demultiplexer in the unbroken $mathcal{PT}$-symmetric regime. The length of the proposed system is of the order of millimeters and the lasing spectra in the broken $mathcal{PT}$-symmetric regime can be easily tuned in terms of intensity, bandwidth and wavelength by varying the magnitude of the phase shift in the middle of the structure. Surprisingly, the multi-modal lasing spectra are suppressed by virtue of judiciously selected phase and the gain-loss value. Also, it is possible to obtain sidelobe-less spectra in the broken $mathcal{PT}$-symmetric regime, without a need for an apodization profile, which is a traditional way to tame the unwanted sidelobes. The system is found to show narrow band single-mode lasing behavior for a wide range of phase shift values for given values of gain and loss. Moreover, we report the intensity tunable reflection and transmission characteristics in the unbroken regime via variation in gain and loss. At the exceptional point, the system shows unidirectional wave transport phenomenon independent of the presence of phase shift in the middle of the grating. For the right light incidence direction, the system exhibits zero reflection wavelengths within the stopband at the exceptional point. We also investigate the role of multiple phase shifts placed at fixed locations along the length of the FBG and the variations in the spectra when the phase shift and gain-loss values are tuned. In the broken $mathcal{PT}$-symmetric regime, the presence of multiple phase shifts aids in controlling the number of reflectivity peaks besides controlling their magnitude.
We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective) quasi-isometric embeddings between relatively hyperbolic groups/spaces that coarsely respect peripherals, and quasisymmetric embeddings between their boundaries satisfying suitable conditions. Further, we establish a similar correspondence regarding maps with at most polynomial distortion. We use this to characterise groups which are hyperbolic relative to some collection of virtually nilpotent subgroups as exactly those groups which admit an embedding into a truncated real hyperbolic space with at most polynomial distortion, generalising a result of Bonk and Schramm for hyperbolic groups.
We define a notion of tracial $mathcal{Z}$-absorption for simple not necessarily unital C*-algebras. This extends the notion defined by Hirshberg and Orovitz for unital (simple) C*-algebras. We provide examples which show that tracially $mathcal{Z}$-absorbing C*-algebras need not be $mathcal{Z}$-absorbing. We show that tracial $mathcal{Z}$-absorption passes to hereditary C*-subalgebras, direct limits, matrix algebras, minimal tensor products with arbitrary simple C*-algebras. We find sufficient conditions for a simple, separable, tracially $mathcal{Z}$-absorbing C*-algebra to be $mathcal{Z}$-absorbing. We also study the Cuntz semigroup of a simple tracially $mathcal{Z}$-absorbing C*-algebra and prove that it is almost unperforated and weakly almost divisible.
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