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Register a new userOn constructions preserving the asymptotic topology of metric spaces
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We prove that graph products constructed over infinite graphs with bounded clique number preserve finite asymptotic dimension. We also study the extent to which Dranishnikovs property C, and Dranishnikov and Zarichnyis straight finite decomposition complexity are preserved by constructions such as unions, free products, and group extensions.
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Let $X_{0}$ be a complete hyperbolic surface of infinite type with geodesic boundary which admits a countable pair of pants decomposition. As an application of the Basmajian identity for complete bordered hyperbolic surfaces of infinite type with limit sets of 1-dimensional measure zero, we define an asymmetric metric (which is called arc metric) on the quasiconformal Teichmuller space $mathcal{T}(X_{0})$ provided that $X_{0}$ satisfies a geometric condition. Furthermore, we construct several examples of hyperbolic surfaces of infinite type satisfying the geometric condition and discuss the relation between the Shigas condition and the geometric condition.
We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the hyperbolic directions in that space. This boundary is a quasi-isometry invariant and thus produces a well-defined boundary for any finitely generated group. In the case of a proper $mathrm{CAT}(0)$ space this boundary is the contracting boundary of Charney and Sultan and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. We prove three results about the Morse boundary of Teichmuller space. First, we show that the Morse boundary of the mapping class group of a surface is homeomorphic to the Morse boundary of the Teichmuller space of that surface. Second, using a result of Leininger and Schleimer, we show that Morse boundaries of Teichmuller space can contain spheres of arbitrarily high dimension. Finally, we show that there is an injective continuous map of the Morse boundary of Teichmuller space into the Thurston compactification of Teichmuller space by projective measured foliations.
We prove that the deformation space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with incompressible boundary is locally connected at minimally parabolic points. Moreover, spaces of Kleinian surface groups are locally connected at quasiconformally rigid points. Similar results are obtained for deformation spaces of acylindrical 3-manifolds and Bers slices.
Property A was introduced by Yu as a non-equivariant analogue of amenability. Nigel Higson posed the question of whether there is a homological characterisation of property A. In this paper we answer Higsons question affirmatively by constructing analogues of group cohomology and bounded cohomology for a metric space X, and show that property A is equivalent to vanishing cohomology. Using these cohomology theories we also give a characterisation of property A in terms of the existence of an asymptotically invariant mean on the space.
Reconstructing Portal Vein and Hepatic Vein trees from contrast enhanced abdominal CT scans is a prerequisite for preoperative liver surgery simulation. Existing deep learning based methods treat vascular tree reconstruction as a semantic segmentation problem. However, vessels such as hepatic and portal vein look very similar locally and need to be traced to their source for robust label assignment. Therefore, semantic segmentation by looking at local 3D patch results in noisy misclassifications. To tackle this, we propose a novel multi-task deep learning architecture for vessel tree reconstruction. The network architecture simultaneously solves the task of detecting voxels on vascular centerlines (i.e. nodes) and estimates connectivity between center-voxels (edges) in the tree structure to be reconstructed. Further, we propose a novel connectivity metric which considers both inter-class distance and intra-class topological distance between center-voxel pairs. Vascular trees are reconstructed starting from the vessel source using the learned connectivity metric using the shortest path tree algorithm. A thorough evaluation on public IRCAD dataset shows that the proposed method considerably outperforms existing semantic segmentation based methods. To the best of our knowledge, this is the first deep learning based approach which learns multi-label tree structure connectivity from images.
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