Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAmenable actions, invariant means and bounded cohomology
252
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We show that topological amenability of an action of a countable discrete group on a compact space is equivalent to the existence of an invariant mean for the action. We prove also that this is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, as well as to vanishing of a specific cohomology class. In the case when the compact space is a point our result reduces to a classic theorem of B.E. Johnson characterising amenability of groups. In the case when the compact space is the Stone-v{C}ech compactification of the group we obtain a cohomological characterisation of exactness for the group, answering a question of Higson.
rate research
Read More
Let $G$ be an amenable group. We define and study an algebra $mathcal{A}_{sn}(G)$, which is related to invariant means on the subnormal subgroups of $G$. For a just infinite amenable group $G$, we show that $mathcal{A}_{sn}(G)$ is nilpotent if and only if $G$ is not a branch group, and in the case that it is nilpotent we determine the index of nilpotence. We next study $operatorname{rad} ell^1(G)^{**}$ for an amenable branch group $G$, and show that it always contains nilpotent left ideals of arbitrarily large index, as well as non-nilpotent elements. This provides infinitely many finitely-generated counterexamples to a question of Dales and Lau, first resolved by the author in a previous article, which asks whether we always have $(operatorname{rad} ell^1(G)^{**})^{Box 2} = { 0 }$. We further study this question by showing that $(operatorname{rad} ell^1(G)^{**})^{Box 2} = { 0 }$ imposes certain structural constraints on the group $G$.
Given a group acting on a Gromov hyperbolic space, Bestvina and Fujiwara introduced the WPD property --- weak proper discontinuity --- for studying the 2nd bounded cohomology of the group. We carry out a more general study of second bounded cohomology using a really weak property discontinuity property known as WWPD that was introduced by Bestvina, Bromberg, and Fujiwara.
We give a new perspective on the homological characterisations of amenability given by Johnson in the context of bounded cohomology and by Block and Weinberger in the context of uniformly finite homology. We examine the interaction between their theories and explain the relationship between these characterisations. We apply these ideas to give a new proof of non- vanishing for the bounded cohomology of a free group.
In this two part work we prove that for every finitely generated subgroup $Gamma < text{Out}(F_n)$, either $Gamma$ is virtually abelian or $H^2_b(Gamma;mathbb{R})$ contains an embedding of $ell^1$. The method uses actions on hyperbolic spaces, for purposes of constructing quasimorphisms. Here in Part I, after presenting the general theory, we focus on the case of infinite lamination subgroups $Gamma$ - those for which the set of all attracting laminations of all elements of $Gamma$ is infinite - using actions on free splitting complexes of free groups.
This is the second part of a two part work in which we prove that for every finitely generated subgroup $Gamma < mathsf{Out}(F_n)$, either $Gamma$ is virtually abelian or its second bounded cohomology $H^2_b(Gamma;mathbb{R})$ contains an embedding of $ell^1$. Here in Part II we focus on finite lamination subgroups $Gamma$ --- meaning that the set of all attracting laminations of elements of $Gamma$ is finite --- and on the construction of hyperbolic actions of those subgroups to which the general theory of Part I is applicable.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
