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Yves de Cornulier

Publications and source records attributed to Yves de Cornulier.

At least 19 recordsLinked to original sources
arXiv2020

On groups with weak Sierpiński subsets

In a group $G$, a weak Sierpiński subset is a subset $E$ such that for some $g,h\in G$ and $a\neq b\in E$, we have $gE=E\smallsetminus \{a\}$ and $hE=E\smallsetminus \{b\}$. In this setting, we study the subgroup generated by $g$ and $h$, and show that it has a special presentation, namely of the form $G_k=\langle g,h\mid (h^{-1}g)^k\rangle$ unless it is free over $(g,h)$. In addition, in such groups $G_k$, we characterize all weak Sierpiński subsets.

math.GR↗
arXiv2020

Aspects de la géométrie des groupes

This habilitation memoir (in French, submitted in May 2014) is made up of five chapters, each being an introduction to work of the author between 2006 and 2014. The core of the memoir consists of the first three chapters, pertaining to geometric group theory. More precisely, it is concerned with large-scale geometry of groups and especially Lie groups (Chapters 1 and 2), and locally compact hyperbolic groups (Chapter 3). Chapter 4 is concerned with the study of groups through their unitary representations and their isometric actions on Hilbert spaces (Kazhdan and Haagerup properties). Finally, Chapter 5 is more concerned with "structural" group theory, studying, for a given group (discrete, or more generally locally compact), the space of its closed subgroups, or closed normal subgroups. A list of open questions is appended. While the memoir mostly surveys published papers, it also presents some unpublished corollaries in Chapter 1 about asymptotic cones of Lie groups.

math.GR↗
arXiv2020

Groupes pleins-topologiques [d'après Matui, Juschenko, Monod...]

This is the written version of the Bourbaki seminar given in January 2013 and published in 2014 (modulo an additional early reference added subsequently). It describes the first construction of infinite, finitely generated amenable simple groups. It starts with a general study of topological-full groups, along with indications on the first appearances of such groups. Then the theorem is proved, namely that the derived subgroup of the topological-full group of an infinite minimal subshift is both simple and finitely generated (Matui 2006) and amenable (Juschenko-Monod 2013, after being conjectured by Grigorchuk-Medynets). Some other properties of these groups are discussed. None of the numerous subsequent developments since 2013 are mentioned.

math.GR↗
arXiv2019

Commensurating actions of birational groups and groups of pseudo-automorphisms

Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension 1. We import ideas from geometric group theory to prove that a group of birational transformations that satisfies a fixed point property on CAT(0) cubical complexes, for example a discrete countable group with Kazhdan Property (T), is birationally conjugate to a group acting by pseudo-automorphisms on some non-empty Zariski-open subset. We apply this argument to classify groups of birational transformations of surfaces with this fixed point property up to birational conjugacy.

math.AG↗
arXiv2013

Amenable hyperbolic groups

We give a complete characterization of the locally compact groups that are non-elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semi-regular trees acting doubly transitively on the set of ends. As an application, we show that the class of hyperbolic locally compact groups with a cusp-uniform non-uniform lattice, is very restricted.

math.GR↗
arXiv2013

Infinite presentability of groups and condensation

We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that a group is infinitely independently presentable. In addition, we construct examples of finitely generated groups with no minimal presentation, among them infinitely presented groups with Cantor-Bendixson rank 1, and we prove that every infinitely presented metabelian group is a condensation group.

math.GR↗
arXiv2012

Finitely presentable, non-Hopfian groups with Kazhdan's Property (T) and infinite outer automorphism group

We give simple examples of Kazhdan groups with infinite outer automorphism groups. This answers a question of Paulin, independently answered by Ollivier and Wise by completely different methods. As arithmetic lattices in (non-semisimple) Lie groups, our examples are in addition finitely presented. We also use results of Abels about compact presentability of p-adic groups to exhibit a finitely presented non-Hopfian Kazhdan group. This answers a question of Ollivier and Wise.

math.GR↗
arXiv2012

Semisimple Zariski closure of Coxeter groups

Let W be an irreducible, finitely generated Coxeter group. The geometric representation provides an discrete embedding in the orthogonal group of the so-called Tits form. One can look at the representation modulo the kernel of this form; we give a new proof of the following result of Vinberg: if W is non-affine, then this representation remains faithful. Our proof uses relative Kazhdan Property (T). The following corollary was only known to hold when the Tits form is non-degenerate: the reduced C*-algebra of W is simple with a unique normalized trace. Some other corollaries are pointed out.

math.GR↗
arXiv2011

On conjugacy growth of linear groups

We investigate the conjugacy growth of finitely generated linear groups. We show that finitely generated non-virtually-solvable subgroups of GL_d have uniform exponential conjugacy growth and in fact that the number of distinct polynomials arising as characteristic polynomials of the elements of the ball of radius n for the word metric has exponential growth rate bounded away from 0 in terms of the dimension d only.

math.GR↗
arXiv2010

The space of subgroups of an abelian group

We carry out the Cantor-Bendixson analysis of the space of all subgroups of any countable abelian group and we deduce a complete classification of such spaces up to homeomorphism.

math.GR↗
arXiv2009

On the isolated points in the space of groups

We investigate the isolated points in the space of finitely generated groups. We give a workable characterization of isolated groups and study their hereditary properties. Various examples of groups are shown to yield isolated groups. We also discuss a connection between isolated groups and solvability of the word problem.

math.GR↗
arXiv2007

Dimension of asymptotic cones of Lie groups

We compute the covering dimension the asymptotic cone of a connected Lie group. For simply connected solvable Lie groups, this is the codimension of the exponential radical. As an application of the proof, we give a characterization of connected Lie groups that quasi-isometrically embed into a non-positively curved metric space.

math.GR↗
arXiv2006

Isometric group actions on Banach spaces and representations vanishing at infinity

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 $({\BP}_0^V)$ if every affine isometric action of $G$ on $V$, such that the linear part is a $C_0$-representation of $G$, either has a fixed point or is metrically proper. We prove that solvable groups, connected Lie groups, and linear algebraic groups over a local field of characteristic zero, have property $({\BP}_0^V)$. As a consequence for unitary representations, we characterize those groups in the latter classes for which the first cohomology with respect to the left regular representation on $L^2(G)$ is non-zero; and we characterize uniform lattices in those groups for which the first $L^2$-Betti number is non-zero.

math.RT↗
arXiv2006

Isometric group actions on Hilbert spaces: growth of cocycles

We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing polycyclic groups and connected amenable Lie groups: either G has no quasi-isometric embedding into Hilbert space, or G admits a proper cocompact action on some Euclidean space. On the other hand, noting that almost coboundaries (i.e. 1-cocycles approximable by bounded 1-cocycles) have sublinear growth, we discuss the converse, which turns out to hold for amenable groups with "controlled" Folner sequences; for general amenable groups we prove the weaker result that 1-cocycles with sufficiently small growth are almost coboundaries. Besides, we show that there exist, on a-T-menable groups, proper cocycles with arbitrary small growth.

math.GR↗