arXiv subjects

Dapeng Zhou

Publications and source records attributed to Dapeng Zhou.

6 recordsLinked to original sources

arXiv2024

Dynamic asymptotic dimension growth for group actions and groupoids

We introduce the notion of dynamic asymptotic dimension growth for actions of discrete groups on compact spaces, and more generally for locally compact \'etale groupoids. Moreover, we demonstrate that the asymptotic dimension growth for a discrete metric space of bounded geometry is equivalent to the dynamic asymptotic dimension growth for its associated coarse groupoid. Consequently, we deduce that the coarse groupoid with subexponential dynamic asymptotic dimension growth is amenable. More generally, we show that every $\sigma$-compact locally compact Hausdorff \'etale groupoid with compact unit space and dynamic asymptotic dimension growth at most $x^{\alpha}$ $(0<\alpha<1)$ is amenable. As an application, we show that the Baum-Connes conjecture with coefficients holds for such groupoids.

math.DS↗

arXiv2023

$L^p$ coarse Baum-Connes conjecture via $C_{0}$ coarse geometry

In this paper, we investigate the $L^{p}$ coarse Baum-Connes conjecture for $p\in [1,\infty)$ via $C_{0}$ coarse structure, which is a refinement of the bounded coarse structure on a metric space. We prove that the $C_{0}$ version of the $L^{p}$ coarse Baum-Connes conjecture holds for a finite-dimensional simplicial complex equipped with a uniform spherical metric. Using this result, we construct an obstruction group for the $L^{p}$ coarse Baum-Connes conjecture. As an application, we show that the obstruction group vanishes under the assumption of finite asymptotic dimension, thereby providing a new proof of the $L^{p}$ coarse Baum-Connes conjecture in this case.

math.FA↗

arXiv2022

Persistence approximation property for $L^p$ operator algebras

In this paper, we study the persistence approximation property for quantitative $K$-theory of filtered $L^p$ operator algebras. Moreover, we define quantitative assembly maps for $L^p$ operator algebras when $p\in [1,\infty)$. Finally, in the case of $L^{p}$ crossed products and $L^{p}$ Roe algebras, we find sufficient conditions for the persistence approximation property. This allows us to give some applications involving the $L^{p}$ (coarse) Baum-Connes conjecture.

math.OA↗

arXiv2021

Uniformly bounded fibred coarse embeddability and uniformly bounded a-T-menability

In this paper, we introduce the concept of uniformly bounded fibred coarse embeddability of metric spaces, generalizing the notion of fibred coarse embeddability defined by X. Chen, Q. Wang and G. Yu. Moreover, we show its relationship with uniformly bounded a-T-menability of groups. Finally, we give some examples to illustrate the differences between uniformly bounded fibred coarse embeddability and fibred coarse embeddability.

math.FA↗

arXiv2021

$L^p$ Coarse Baum-Connes Conjecture and $K$-theory for $L^p$ Roe Algebras

In this paper, we verify the $L^p$ coarse Baum-Connes conjecture for spaces with finite asymptotic dimension for $p\in[1,\infty)$. We also show that the $K$-theory of $L^p$ Roe algebras are independent of $p\in(1,\infty)$ for spaces with finite asymptotic dimension.

math.KT↗

arXiv2020

Relative Equivariant Coarse Index Theorem and Relative $L^2$-Index Theorem

In this paper, we give a definition of the relative equivariant coarse index for proper actions and derive a relative equivariant coarse index theorem connecting this index with the localized equivariant coarse indices. This is an equivariant version of Roe's relative coarse index theorem in arXiv:arch-ive/1210.6100. Furthermore, we present a definition of the relative $L^2$-index and prove a relative $L^2$-index theorem which is a relative version of Atiyah's $L^2$-index theorem.

math.OA↗