Group approximation in Cayley topology and coarse geometry, III: Geometric property (T)

This paper studies higher index theory for a random sequence of bounded degree, finite graphs with diameter tending to infinity. We show that in a natural model for such random sequences the following hold almost surely: the coarse Baum–Connes assembly map is injective; the coarse Baum–Connes assembly map is not surjective; the maximal coarse Baum–Connes assembly map is an isomorphism. These results are closely tied to issues of expansion in graphs: in particular, we also show that such random sequences almost surely do not have geometric property (T), a strong form of expansion.The key geometric ingredients in the proof are due to Mendel and Naor: in our context, their results imply that a random sequence of graphs almost surely admits a weak form of coarse embedding into Hilbert space.

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Geometric property (T)

Chinese Annals of Mathematics Series B ◽

10.1007/s11401-014-0852-x ◽

2014 ◽

Vol 35 (5) ◽

pp. 761-800 ◽

Cited By ~ 7

Author(s):

Rufus Willett ◽

Guoliang Yu

Keyword(s):

Geometric Property ◽

Property T

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Geometric property (T) for non-discrete spaces

Journal of Functional Analysis ◽

10.1016/j.jfa.2021.109148 ◽

2021 ◽

pp. 109148

Author(s):

Jeroen Winkel

Keyword(s):

Geometric Property ◽

Property T ◽

Discrete Spaces

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Geometric property-based convolutional neural network for indoor object detection

International Journal of Advanced Robotic Systems ◽

10.1177/1729881421993323 ◽

2021 ◽

Vol 18 (1) ◽

pp. 172988142199332

Author(s):

Xintao Ding ◽

Boquan Li ◽

Jinbao Wang

Keyword(s):

Neural Network ◽

Object Detection ◽

Convolutional Neural Network ◽

Geometric Property ◽

Ground Truth ◽

Geometric Constraints ◽

Depth Information ◽

Training Set ◽

Object Knowledge ◽

The Mean

Indoor object detection is a very demanding and important task for robot applications. Object knowledge, such as two-dimensional (2D) shape and depth information, may be helpful for detection. In this article, we focus on region-based convolutional neural network (CNN) detector and propose a geometric property-based Faster R-CNN method (GP-Faster) for indoor object detection. GP-Faster incorporates geometric property in Faster R-CNN to improve the detection performance. In detail, we first use mesh grids that are the intersections of direct and inverse proportion functions to generate appropriate anchors for indoor objects. After the anchors are regressed to the regions of interest produced by a region proposal network (RPN-RoIs), we then use 2D geometric constraints to refine the RPN-RoIs, in which the 2D constraint of every classification is a convex hull region enclosing the width and height coordinates of the ground-truth boxes on the training set. Comparison experiments are implemented on two indoor datasets SUN2012 and NYUv2. Since the depth information is available in NYUv2, we involve depth constraints in GP-Faster and propose 3D geometric property-based Faster R-CNN (DGP-Faster) on NYUv2. The experimental results show that both GP-Faster and DGP-Faster increase the performance of the mean average precision.

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Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

International Journal of Mathematics ◽

10.1142/s0129167x15500640 ◽

2015 ◽

Vol 26 (08) ◽

pp. 1550064

Author(s):

Bachir Bekka

Keyword(s):

Von Neumann Algebra ◽

Regular Representation ◽

Von Neumann ◽

Lp Spaces ◽

Local Rigidity ◽

Left Regular Representation ◽

Left Regular ◽

Property T ◽

Finite Factor ◽

Finite Index Subgroup

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

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