Group actions on Banach spaces and a geometric characterization of a-T-menability

Geometric Property (T)

This paper discusses `geometric property (T)'. This is a property of metric spaces introduced in earlier work of the authors for its applications to K-theory. Geometric property (T) is a strong form of `expansion property': in particular for a sequence of finite graphs $(X_n)$, it is strictly stronger than $(X_n)$ being an expander in the sense that the Cheeger constants $h(X_n)$ are bounded below. We show here that geometric property (T) is a coarse invariant, i.e. depends only on the large-scale geometry of a metric space $X$. We also discuss the relationships between geometric property (T) …

通过 Roe 代数的 Banach 表示研究度量空间的几何 Banach 性质 (T)

Geometric Banach property (T) for metric spaces via Banach representations of Roe algebras

本文通过 Roe 代数的 Banach 表示引入度量空间的几何 Banach 性质 (T)，证明其是粗几何不变量并等价于 Kazhdan 投影存在，同时研究其对有限 Cayley 图序列及剩余有限群强 Banach 性质 (T) 的刻画。

When symplectic topology meets Banach space geometry

In this paper we survey some recent works that take the first steps toward establishing bilateral connections between symplectic geometry and several other fields, namely, asymptotic geometric analysis, classical convex geometry, and the theory of normed spaces.

Property $(β)$ and uniform quotient maps

In 1999, Bates, Johnson, Lindenstrauss, Preiss and Schechtman asked whether a Banach space that is a uniform quotient of $\ell_p$, $1 < p \neq 2 < \infty$, must be isomorphic to a linear quotient of $\ell_p$. We apply the geometric property $(β)$ of Rolewicz to the study of uniform and Lipschitz quotient maps, and answer the above question positively for the case $1<p<2$. We also give a necessary condition for a Banach space to have $c_0$ as a uniform quotient.

Geometric rough paths on infinite dimensional spaces

Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For $α\in (1/3,1/2)$, we give criteria for when we can approximate Banach space-valued weakly geometric $α$-rough paths by signatures of curves of bounded variation, given some tuning of the Hölder parameter. We show that these criteria are satisfied for weakly geometric rough paths on Hilbert spaces. As an application, we obtain Wong-Zakai type result for function space valued martingales using the notion of (unbounded) rough drivers.

群的索菲边界与索菲近似的大尺度几何

Sofic boundaries of groups and coarse geometry of sofic approximations

研究通过索菲近似定义群的索菲边界，证明大尺度性质如属性A、Hilbert空间渐近嵌入性和几何属性(T)对应于群的分析性质，扩展了残余有限群的研究成果。