Related papers: Banach space actions and $L^2$-spectral gap

We prove that a random group in the triangular density model has, for density larger than 1/3, fixed point properties for actions on $L^p$-spaces (affine isometric, and more generally $(2-2\epsilon)^{1/2p}$-uniformly Lipschitz) with $p$…

We prove that all isometric actions of higher rank simple Lie groups and their lattices on arbitrary uniformly convex Banach spaces have a fixed point. This vastly generalises a recent breakthrough of Oppenheim. Combined with earlier work…

We show that every non-precompact topological group admits a fixed point-free continuous action by affine isometries on a suitable Banach space. Thus, precompact groups are defined by the fixed point property for affine isometric actions on…

Our main result is that the simple Lie group $G=Sp(n,1)$ acts properly isometrically on $L^p(G)$ if $p>4n+2$. To prove this, we introduce property $({\BP}_0^V)$, for $V$ be a Banach space: a locally compact group $G$ has property…

The objective of this series is to study metric geometric properties of disjoint unions of amenable Cayley graphs by group properties of the Cayley accumulation points in the space of marked groups. In this Part II, we prove that a disjoint…

We prove a geometric characterization of a-T-menability through proper, affine, isometric actions on the Banach spaces $L_p[0,1]$ for $1<p<2$. This answers a question of A. Valette.

For a topological group $G$, amenability can be characterized by the amenability of the convolution Banach algebra $L^1(G)$. Here a Banach algebra $A$ is called amenable if every bounded derivation from $A$ into any dual--type…

The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group $G$ on a reflexive…

We show that for any group $G$ that is hyperbolic relative to subgroups that admit a proper affine isometric action on a uniformly convex Banach space, then $G$ acts properly on a uniformly convex Banach space as well.

We consider certain strengthenings of property (T) relative to Banach spaces that are satisfied by high rank Lie groups. Let X be a Banach space for which, for all k, the Banach--Mazur distance to a Hilbert space of all k-dimensional…

We prove a number of results linking properties of actions by compact groups (both quantum and classical) on Banach spaces, such as uniform continuity, spectrum finiteness and extensibility of the actions across several constructions.…

We prove that any hyperbolic group admits a proper affine isometric action on a quotient space of a $\ell^p$ Banach space, for all $p>1$ sufficiently close to 1.

In [Laf08], [Laf09], Vincent Lafforgue proved strong Banach property (T) for $SL_3$ over a non archimedean local field $F.$ In this paper, we extend his results to $Sp_4$ and therefore to any connected almost $F$-simple algebraic group with…

We show that the lamplighter group L has a system of generators for which the spectrum of the discrete Laplacian on the Cayley graph is a union of an interval and a countable set of isolated points accumulating to a point outside this…

We show that every metric space with bounded geometry uniformly embeds into an explicit reflexive Banach space (a direct sum of l^p spaces). In the case of discrete groups we show the analogue of a-T-menability. That is, we construct a…

In this paper we consider bounded operators on infinite graphs, in particular Cayley graphs of amenable groups. The operators satisfy an equivariance condition which is formulated in terms of a colouring of the vertex set of the underlying…

The fixed-point spectrum of a locally compact second countable group G on lp is defined to be the set of real numbers p such that every action by affine isometries of G on lp admits a fixed-point. We show that this set is either empty, or…

In this paper, we deal with a notion of Banach space-valued mappings defined on a set consisting of finite graphs with uniformly bounded vertex degree. These functions will be endowed with certain boundedness and additivity criteria. We…

We study property (T) and the fixed point property for actions on $L^p$ and other Banach spaces. We show that property (T) holds when $L^2$ is replaced by $L^p$ (and even a subspace/quotient of $L^p$), and that in fact it is independent of…

In this paper, we give a necessary and sufficient condition for which a finitely generated group has a property like Kazhdan's Property $(T)$ restricted to one isometric representation on a strictly convex Banach space without non-zero…