Searcharxiv⌕ Search

arXiv subjects

Ignacio Vergara

Publications and source records attributed to Ignacio Vergara.

At least 19 recordsLinked to original sources
arXiv2026

Relative property (T) for uniformly bounded representations

For pairs $(G,H)$, where $G$ is a locally compact group and $H$ is a closed subgroup of $G$, we introduce analogous versions of the full group $\mathrm{C}^*$-algebra and the Fourier--Stieltjes algebra for uniformly bounded representations. We use these objects to characterise relative property (T) in this more general setting in terms of Kazhdan projections and invariant means. We also show that, for semidirect products of the form $G=H\ltimes N$, where $N$ is a nilpotent group, relative property (T) for the pair $(G,N)$ is equivalent to its uniformly bounded version. We first prove this result when $N$ is abelian, and then proceed by studying the behaviour of relative property (T) under quotients by central subgroups, thereby extending a theorem of Serre to the uniformly bounded setting.

math.GR↗
arXiv2026

$L_1$ Actions and Embeddings of Property A Spaces

We provide several new characterizations of Property A for bounded degree graphs. In particular, we show that $(X,d)$ has Property A if and only if there is a proper gauge $\omega$ such that the Lipschitz free space $\operatorname{LF}(X,\omega\circ d)$ is isomorphic to $\ell_1$. As a consequence, all finitely generated groups with Property A admit proper uniformly Lipschitz affine actions on $\ell_1$. Moreover, for groups with finite Nagata dimension, we obtain actions with compression exponent 1. This result applies to higher rank lattices, such as $\operatorname{SL}(3,\mathbb{Z})$. We also show that a countable discrete group coarsely embeds into $L_1$ if and only if it admits a proper uniformly Lipschitz affine action on a subspace of $L_1$.

math.GR↗
arXiv2025

Some remarks on $M_d$-multipliers and approximation properties

We prove an extension property for $M_d$-multipliers from a subgroup to the ambient group, showing that $M_{d+1}(G)$ is strictly contained in $M_d(G)$ whenever $G$ contains a free subgroup. Another consequence of this result is the stability of the $M_d$-approximation property under group extensions. We also show that Baumslag-Solitar groups are $M_d$-weakly amenable with $\boldsymbol\Lambda(\operatorname{BS}(m,n),d)=1$ for all $d\geq 2$. Finally, we show that, for simple Lie groups with finite centre, $M_d$-weak amenability is equivalent to weak amenability, and we provide some estimates on the constants $\boldsymbol\Lambda(G,d)$.

math.GR↗
arXiv2024

Quasi-trees, Lipschitz free spaces, and actions on $\ell^1$

We show that the Lipschitz free space of a countable simplicial quasi-tree is isomorphic to $\ell^1$. As a consequence, every finitely generated group with Property (QT) of Bestvina--Bromberg--Fujiwara has a proper uniformly Lipschitz affine action on $\ell^1$ with quasi-isometrically embedded orbits. We also show that $3$-manifold groups admit proper uniformly Lipschitz affine actions on $\ell^1$.

math.GR↗
arXiv2024

A connection between Lipschitz and Kazhdan constants for groups of homeomorphisms of the real line

We exhibit an obstruction for groups with Relative Property (T) to act on the real line by bi-Lipschitz homeomorphisms. This condition is expressed in terms of the Lipschitz and Kazhdan constants associated to finite generating subsets. As an application, we obtain an explicit lower bound for the Lipschitz constants associated to actions of the semidirect product $\mathbb{F}_2\ltimes\mathbb{Z}^2$. We also obtain an upper bound for the Kazhdan constants of pairs of orderable groups, depending only on the cardinal of the generating subset.

math.GR↗
arXiv2024

Geometric Property (T) and Kazhdan projections

We characterise Geometric Property (T) by the existence of a certain projection in the maximal uniform Roe algebra $C_{u,\max}^*(X)$, extending the notion of Kazhdan projection for groups to the realm of metric spaces. We also describe this projection in terms of the decomposition of the metric space into coarsely connected components.

math.OA↗
arXiv2024

Property $\mathrm{(TTT)}$ and the Cowling--Haagerup constant

We show that Property $\mathrm{(TTT)}$ is an obstruction to weak amenability with Cowling--Haagerup constant $1$. More precisely, if $G$ is a countable group and $H$ is an infinite subgroup of $G$ such that the pair $(G,H)$ has relative Property $\mathrm{(TTT)}$, then the weak Haagerup constant $\boldsymbol\Lambda_{\mathrm{WH}}(G)$ is strictly greater than $1$. We apply this result to some semidirect products and lattices in higher rank algebraic groups.

math.GR↗
arXiv2023

Property (T) for uniformly bounded representations and weak*-continuity of invariant means

For every $c\geq 1$, we define a strengthening of Kazhdan's Property (T) by considering uniformly bounded representations $π$ with fixed bound $|π|\leq c$. We carry out a systematic study of this property and show that it can be characterised by the weak*-continuity of the unique invariant mean on a suitable space of coefficients. For countable groups, we prove that the family of properties thus obtained yield an invariant at the von Neumann algebra level. Moreover, by focusing on certain representations of rank 1 Lie groups, we show that $\operatorname{Sp}(n,1)$ and $F_{4,-20}$ admit proper uniformly Lipschitz affine actions on Hilbert spaces.

math.GR↗
arXiv2023

Hyperbolicity and uniformly Lipschitz affine actions on subspaces of $L^1$

We show that every hyperbolic group has a proper uniformly Lipschitz affine action on a subspace of an $L^1$ space. We also prove that every acylindrically hyperbolic group has a uniformly Lipschitz affine action on such a space with unbounded orbits. Our main tools are the $\mathbb{Q}$-bicombings on hyperbolic groups constructed by Mineyev and the characterisation of acylindrical hyperbolicity in terms of actions on quasi-trees by Balasubramanya.

math.GR↗
arXiv2022

The $M_d$-Approximation Property and Unitarisability

We define a strengthening of the Haagerup-Kraus approximation property by means of the subalgebras of Herz-Schur multipliers $M_d(G)$ ($d\geq 2$) introduced by Pisier. We show that unitarisable groups satisfying this property for all $d\geq 2$ are amenable. Moreover, we show that groups acting properly on finite-dimensional CAT(0) cube complexes satisfy $M_d$-AP for all $d\geq 2$. We also give examples of non-weakly amenable groups satisfying $M_d$-AP for all $d\geq 2$.

math.GR↗
arXiv2021

Almost invariant CND kernels and proper uniformly Lipschitz actions on subspaces of $L^1$

We define the notion of almost invariant conditionally negative definite kernel and use it to give a characterisation of groups admitting a proper uniformly Lipschitz affine action on a subspace of an $L^1$ space. We show that this condition is satisfied by groups acting properly on products of quasi-trees, weakly amenable groups with Cowling-Haagerup constant 1, and a-TTT-menable groups.

math.GR↗
arXiv2019

Positive definite radial kernels on homogeneous trees and products

We give a new proof of a classical result which provides a one-to-one correspondence between positive definite radial kernels on a homogeneous tree and finite Borel measures on the interval $[-1,1]$. Our methods allow us to find a new characterisation in terms of positive trace-class operators on $\ell_2$. Furthermore, we extend both characterisations to finite products of homogeneous trees. The proof relies on a formula for the norm of radial Schur multipliers, in the spirit of Haagerup--Steenstrup--Szwarc, and a variation of the Hamburger moment problem.

math.FA↗
arXiv2018

Radial Schur multipliers on some generalisations of trees

We give a characterisation of radial Schur multipliers on finite products of trees. The equivalent condition is that a certain generalised Hankel matrix involving the discrete derivatives of the radial function is a trace class operator. This extends Haagerup, Steenstrup and Szwarc's result for trees. The same condition can be expressed in terms of Besov spaces on the torus. We also prove a similar result for products of hyperbolic graphs and provide a sufficient condition for a function to define a radial Schur multiplier on a finite dimensional CAT(0) cube complex.

math.OA↗
arXiv2017

The $p\,$-approximation property for simple Lie groups with finite center

We prove that, for any $1<p<\infty$, the groups $\text{SL}(3,\mathbb{R})$ and $\text{Sp}(2,\mathbb{R})$ do not have the $p\,$-approximation property of An, Lee and Ruan, which implies in particular that they are not $p\,$-weakly amenable. It follows that the same holds for any connected simple Lie group with finite center and real rank greater than 1, as well as for any lattice in it. This extends Haagerup and de Laat's result for the AP, which in this language corresponds to the case $p=2$.

math.FA↗
arXiv2015

Phase separation in the non-equilibrium Verwey transition in magnetite

We present equilibrium and out-of-equilibrium studies of the Verwey transition in magnetite. In the equilibrium optical conductivity, we find a step-like change at the phase transition for photon energies below about 2 eV. The possibility of triggering a non-equilibrium transient metallic state in insulating magnetite by photo excitation was recently demonstrated by an x-ray study. Here we report a full characterization of the optical properties in the visible frequency range across the non-equilibrium phase transition. Our analysis of the spectral features is based on a detailed description of the equilibrium properties. The out-of-equilibrium optical data bear the initial electronic response associated to localized photo-excitation, the occurrence of phase separation, and the transition to a transient metallic phase for excitation density larger than a critical value. This allows us to identify the electronic nature of the transient state, to unveil the phase transition dynamics, and to study the consequences of phase separation on the reflectivity, suggesting a spectroscopic feature that may be generally linked to out-of-equilibrium phase separation.

cond-mat.str-el↗
arXiv2014

Europium Underneath Graphene on Ir(111): Intercalation Mechanism, Magnetism, and Band Structure

The intercalation of Eu underneath Gr on Ir(111) is comprehensively investigated by microscopic, magnetic, and spectroscopic measurements, as well as by density functional theory. Depending on the coverage, the intercalated Eu atoms form either a $(2 \times 2)$ or a $(\sqrt{3} \times \sqrt{3})$R$30^{\circ}$ superstructure with respect to Gr. We investigate the mechanisms of Eu penetration through a nominally closed Gr sheet and measure the electronic structures and magnetic properties of the two intercalation systems. Their electronic structures are rather similar. Compared to Gr on Ir(111), the Gr bands in both systems are essentially rigidly shifted to larger binding energies resulting in n-doping. The hybridization of the Ir surface state $S_1$ with Gr states is lifted, and the moire superperiodic potential is strongly reduced. In contrast, the magnetic behavior of the two intercalation systems differs substantially as found by X-ray magnetic circular dichroism. The $(2 \times 2)$ Eu structure displays plain paramagnetic behavior, whereas for the $(\sqrt{3} \times \sqrt{3})$R$30^{\circ}$ structure the large zero-field susceptibility indicates ferromagnetic coupling, despite the absence of hysteresis at 10 K. For the latter structure, a considerable easy-plane magnetic anisotropy is observed and interpreted as shape anisotropy.

cond-mat.mes-hall↗