Rapid decay and Metric Approximation Property

Some geometric groups with rapid decay

We explain some simple methods to establish the property of Rapid Decay for a number of groups arising geometrically. We also give new examples of groups with the property of Rapid Decay. In particular we establish the property of Rapid Decay for all lattices in rank one Lie groups.

Rapid decay and Baum-Connes for large type Artin groups

We prove that many Artin groups of large type satisfy the rapid decay property, including all those of extra-large type. For many of these, including all 3-generator groups of extra-large type, a result of Lafforgue applies to show that the groups satisfy the Baum-Connes conjecture without coefficients. Our proof of rapid decay combines elementary analysis with combinatorial techniques, and relies on properties of geodesic words in Artin groups of large type that were observed in an earlier publication by two of the authors of this current article.

The Rapid Decay property and centroids in groups

This is a survey of methods of proving or disproving the Rapid Decay property in groups. We present a centroid property of group actions on metric spaces. That property is a generalized (and corrected) version of the property (**)-relative hyperbolicity" from our paper with Cornelia Drutu, math/0405500, and implies the Rapid Decay (RD) property. We show that several properties which are known to imply RD also imply the centroid property. Thus uniform lattices in many semi-simple Lie groups, Artin groups of large type and the mapping class groups have the centroid property. We also present a si…

Radial rapid decay does not imply rapid decay

We provide a new, dynamical criterion for the radial rapid decay property. We work out in detail the special case of the group $Γ:= \mathbf{SL}_2(A)$, where $A := \mathbb{F}_q[X,X^{-1}]$ is the ring of Laurent polynomials with coefficients in $\mathbb{F}_q$, endowed with the length function coming from a natural action of $Γ$ on a product of two trees, to show that is has the radial rapid decay (RRD) property and doesn't have the rapid decay (RD) property. The criterion also applies to irreducible lattices in semisimple Lie groups with finite center endowed with a length function defined with …

自由群快速衰减性质的Riesz-Thorin方法

A Riesz-Thorin Approach to the Rapid Decay Property for Free Groups

利用Riesz-Thorin插值定理，通过分析自由群在Gromov边界上的拟正则表示，给出Haagerup不等式的新证明，从而建立自由群的快速衰减性质。

Mini-course: property of rapid decay

These are the notes for the mini-course on the property of rapid decay given at the Erwin Schrödinger Institute in Vienna between March 14th and 18th, 2016, as a part of the Measured Group Theory program. They include parts which were omitted during the lectures, and omit some parts which were included in the lectures (and now I think that they shouldn't have been). We explore the definitions of property RD and its permanence properties, discuss the centroid property and relative centroid property defined by Mark Sapir, and give proofs that certain classes of groups have RD.

On the degree of rapid decay

A finitely generated group $\G$ equipped with a word-length is said to satisfy property RD if there are $C, s\geq 0$ such that, for all non-negative integers $n$, we have $\|a\|\leq C (1+n)^s \|a\|_2$ whenever $a\in\C\G$ is supported on elements of length at most $n$. We show that, for infinite $\G$, the degree $s$ is at least 1/2.

On the Hochschild homology of l^1-rapid decay group algebras

We show that for many semi-hyperbolic groups the decomposition into conjugacy classes of the Hochschild homology of the l^1-rapid decay group algebra is injective.

群图上的畸变与三维流形群的快速衰减分类

Distortion in graphs of groups and Rapid Decay classification of 3-manifold groups

本文通过研究群图中的长度函数畸变及快速衰减性质在多项式畸变群图中的稳定性，证明了闭合不可约三维流形的基本群具有快速衰减性质当且仅当它不是虚拟可解群。

Geometry of infinitely presented small cancellation groups, Rapid Decay and quasi-homomorphisms

We study the geometry of infinitely presented groups satisfying the small cancelation condition C'(1/8), and define a standard decomposition (called the criss-cross decomposition) for the elements of such groups. We use it to prove the Rapid Decay property for groups with the stronger small cancelation property C'(1/10). As a consequence, the Metric Approximation Property holds for the reduced C*-algebra and for the Fourier algebra of such groups. Our method further implies that the kernel of the comparison map between the bounded and the usual group cohomology in degree 2 has a basis of power…

Spherical functions and rapid decay for hyperbolic groups

We investigate properties of some spherical fonctions defined on hyperbolic groups using boundary representations on the Gromov boundary endowed with the Patterson-Sullivan measure class. We prove sharp decay estimates for spherical functions as well as spectral inequalities associated with boundary representations. This point of view on the boundary allows us to view the so-called \emph{property RD} (also called \emph{Haagerup's inequality}) as a particular case of a more general behavior of spherical functions on hyperbolic groups. In particular, we give a constructive proof using a boundary…

On groups, slow heat kernel decay yields Liouville property and sharp entropy bounds

Let $μ$ be a symmetric probability measure of finite entropy on a group $G$. We show that if $-\log μ^{(2n)}(id)=o(n^{1/2})$, then the pair $(G,μ)$ has the Liouville property (all bounded $μ$-harmonic functions on $G$ are constant). Furthermore, if $-\log μ^{(2n)}(id)=O(n^β)$ where $β\in(0,1/2)$, then the entropy of the $n$-fold convolution power $μ^{(n)}$ satisfies $H(μ^{(n)})=O\left(n^{\fracβ{1-β}}\right)$. This improves earlier results of Gournay and of Saloff-Coste and the second author. We extend the bounds to transitive graphs and illustrate their sharpness on a family of groups.

熵在齐性空间上的研究以及具有配快速衰减性质的子群分类结果

Entropy on Homogeneous Spaces and Classification Results for Subgroups with the Pair Rapid Decay Property

本文研究了齐性空间$G/H$的配快速衰减性质及其在随机游走和子群结构中的应用，扩展了快速衰减群的熵框架，证明了在$G/H$上的渐近香农熵与特定谱半径量$ c(G,H;μ) $一致，并在某些条件下渐近瑞尼熵率收敛于香农熵。

On the Hochschild and cyclic (co)homology of rapid decay group algebras

We show that the technical condition of solvable conjugacy bound, introduced in \cite{JOR1}, can be removed without affecting the main results of that paper. The result is a Burghelea-type description of the summands $HH_*^t(\BG)_{<x>}$ and $HC_*^t(\BG)_{<x>}$ for any bounding class $\B$, discrete group with word-length $(G,L)$ and conjugacy class $<x>\in <G>$. We use this description to prove the conjecture $\B$-SrBC of \cite{JOR1} for a class of groups that goes well beyond the cases considered in that paper. In particular, we show that the conjecture $\ell^1$-SrBC (the Strong Bass Conjectur…