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Yves Cornulier

Publications and source records attributed to Yves Cornulier.

At least 19 recordsLinked to original sources
arXiv2023

Locally compact modules over abelian groups and compactly generated metabelian groups

We perform a general study of the structure of locally compact modules over compactly generated abelian groups. We obtain a devissage result for such modules of the form "compact-by-sheer-by-discrete", and then study more specifically the sheer part. The main typical example of a sheer module is a polycontractable module, i.e., a finite direct product of modules, each of which is contracted by some group element. We show that every sheer module has a "large" polycontractable submodule, in a suitable sense. We apply this to the study of compactly generated metabelian groups. For instance, we prove that they always have a maximal compact normal subgroup, and we extend the Bieri-Strebel characterization of compactly presentable metabelian groups from the discrete case to this more general setting.

math.GR↗
arXiv2022

On groups of rectangle exchange transformations

We study a generalization Rec_d of the group IET=Rec_1 of interval exchange transformations in every dimension d>0, called the rectangle exchange transformations group. The subset of restricted rotations in IET is a generating subset and we prove that a natural generalization of these elements, called restricted shuffles, form a generating subset of Rec_d. We denote by T_d the subset of Rec_d made up of those transformations that permute two disjoint rectangles by translations. We prove that the derived subgroup of Rec_d is generated by T_d. We also identify the abelianization of Rec_d.

math.GR↗
arXiv2021

Distortion in Cremona groups

We study distortion of elements in two-dimensional Cremona groups over algebraically closed fields of characteristic zero. Namely, we obtain the following trichotomy: non-elliptic elements (i.e., those whose powers have unbounded degree) are undistorted; elliptic elements have a double exponential distortion when they are virtually unipotent and an exponential distortion otherwise.

math.AG↗
arXiv2020

Property FW and 1-dimensional piecewise groups

Property FW is a natural combinatorial weakening of Kazhdan's Property T. We prove that the group of piecewise homographic self-transformations of the real projective line, has "few" infinite subgroups with Property FW. In particular, no such subgroup is amenable or has Kazhdan's Property T. These results are extracted from a longer paper. We provide a complete proof, whose main tools are the use of the notion of partial action and of one-dimensional geometric structures.

math.DS↗
arXiv2020

Irreducible lattices, invariant means, and commensurating actions

We study rigidity properties of lattices in terms of invariant means and commensurating actions (or actions on CAT(0) cube complexes). We notably study Property FM for groups, namely that any action on a discrete set with an invariant mean has a finite orbit.

math.GR↗
arXiv2020

On the quasi-isometric classification of locally compact groups

This (quasi-)survey addresses the quasi-isometry classification of locally compact groups, with an emphasis on amenable hyperbolic locally compact groups. This encompasses the problem of quasi-isometry classification of homogeneous negatively curved manifolds. A main conjecture provides a general description; an extended discussion reduces this conjecture to more specific statements. In the course of the paper, we provide statements of quasi-isometric rigidity for general symmetric spaces of noncompact type and also discuss accessibility issues in the realm of compactly generated locally compact groups.

math.GR↗
arXiv2020

First-order recognisability in finite and pseudofinite groups

It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with 'solubility' replaced by 'nilpotence' and 'perfectness', among others, are false. These facts present difficulties for the study of pseudofinite groups. However, a very weak form of Frattini's theorem on the nilpotence of the Frattini subgroup of a finite group is proved for pseudofinite groups.

math.GR↗
arXiv2019

Realizations of groups of piecewise continuous transformations of the circle

We study the near action of the group PC of piecewise continuous self-transformations of the circle. Elements of this group are only defined modulo indeterminacy on a finite subset, which raises the question of realizability: a subgroup of PC is said to be realizable if it can be lifted to a group of permutations of the circle. We show that every finitely generated abelian subgroup of PC is realizable. We show that this is not true for arbitrary subgroups, by exhibiting a non-realizable finitely generated subgroup of the group of interval exchanges with flips. The group of (oriented) interval exchanges is obviously realizable (choosing the unique left-continuous representative). We show that it has only two realizations (up to conjugation by a finitely supported permutation): the left and right-continuous ones.

math.DS↗
arXiv2019

Counting submodules of a module over a noetherian commutative ring

We count the number of submodules of an arbitrary module over a countable noetherian commutative ring. We give, along the way, a structural description of meager modules, which are defined as those that do not have the square of a simple module as subquotient. We deduce in particular a characterization of uniserial modules over commutative noetherian rings.

math.AC↗
arXiv2019

On the space of ends of infinitely generated groups

We study the space of ends of groups. For a finitely generated group, this is a Cantor space as soon as it is infinite. In contrast, we show that for infinitely generated countable groups, it exhibits several behaviors. For instance, we show that for the free product Z*Q, it is a Cantor space, while for a free group of infinite rank, it is not metrizable. For arbitrary countable groups, we actually establish an alternative: the space of ends is either metrizable, or has a continuous map onto the Stone-Cech compactification of N. We also show that the space of ends of a countable group has a continuous map onto the Stone-Cech boundary of N if and only if the group is infinite locally finite, and that otherwise it is separable. For arbitrary groups, we also prove that the space of ends, if infinite, has no isolated point. We also consider these questions for locally compact groups; for instance we extend Holt's theorem by showing that non-sigma-compact regionally elliptic groups are 1-ended.

math.GR↗
arXiv2019

Near actions

A near permutation of a set is a bijection between two cofinite subsets, modulo coincidence on smaller cofinite subsets. Near permutations of a set form its near symmetric group. In this monograph, we define near actions as homomorphisms into this group, and perform a systematic study of this notion. Notably, we deal with the problem of realizability: how far is a near action from being induced by an action? Also, what invariants of group actions are actually invariants of the underlying near action? the most fundamental one is the space of ends. We notably provide a classification of near actions of finitely generated abelian groups. There are several examples of exotic near actions of the free abelian group of rank two, which we classify according to a positive integer, called winding number, and an integral vector, called additive holonomy. In contrast, we prove that there are no such exotic near actions in higher rank abelian groups. We also obtain results for other groups; for instance, every near action of SL(2,Z) is realizable; this is not true for arbitrary amalgams of two finite cyclic groups. We introduce and study almost and near automorphism groups, which have been considered for some particular graphs or labeled graphs, in the general setting of relational structures.

math.GR↗
arXiv2018

On sublinear bilipschitz equivalence of groups

We discuss the notion of sublinearly bilipschitz equivalences (SBE), which generalize quasi-isometries, allowing some additional terms that behave sublinearly with respect to the distance from the origin. Such maps were originally motivated by the fact they induce bilipschitz homeomorphisms between asymptotic cones. We prove here that for hyperbolic groups, they also induce Hölder homeomorphisms between the boundaries. This yields many basic examples of hyperbolic groups that are pairwise non-SBE. Besides, we check that subexponential growth is an SBE-invariant. The central part of the paper addresses nilpotent groups. While classification up to sublinearly bilipschitz equivalence is known in this case as a consequence of Pansu's theorems, its quantitative version is not. We introduce a computable algebraic invariant $e=e_G<1$ for every such group $G$, and check that $G$ is $O(r^e)$-bilipschitz equivalent to its associated Carnot group. Here $r\mapsto r^e$ is a quantitive sublinear bound. Finally, we define the notion of large-scale contractable and large-scale homothetic metric spaces. We check that these notions imply polynomial growth under general hypotheses, and formulate conjectures about groups with these properties.

math.GR↗
arXiv2018

Locally compact wreath products

Wreath products of non-discrete locally compact groups are usually not locally compact groups, nor even topological groups. We introduce a natural extension of the wreath product construction to the setting of locally compact groups. As an application, we disprove a conjecture of Trofimov, constructing compactly generated locally compact groups of intermediate growth without nontrivial compact normal subgroups.

math.GR↗
arXiv2018

Commensurating actions for groups of piecewise continuous transformations

We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating actions as the existence of a model for which the group acts preserving the geometric structure. We apply this to many groups with piecewise properties in dimension 1, notably piecewise of class C^k, piecewise affine, piecewise projective (possibly discontinuous). We derive various conjugacy results for subgroups with Property FW, or distorted cyclic subgroups, or more generally in the presence of rigidity properties for commensurating actions. For instance we obtain, under suitable assumptions, the conjugacy of a given piecewise affine action to an affine action on possibly another model. By the same method, we obtain a similar result in the projective case. An illustrating corollary is the fact that the group of piecewise projective self-transformations of the circle has no infinite subgroup with Kazhdan's Property T; this corollary is new even in the piecewise affine case. In addition, we use this to provide of the classification of circle subgroups of piecewise projective homeomorphisms of the projective line. The piecewise affine case is a classical result of Minakawa.

math.DS↗
arXiv2017

Commability and focal locally compact groups

We introduce the notion of commability between locally compact groups, namely the equivalence relation generated by cocompact inclusions and quotients by compact normal subgroups. We give a classification of focal hyperbolic locally compact groups up to commability. In the mixed case, it involves a real parameter, which is shown to be a quasi-isometry invariant.

math.GR↗
arXiv2017

On the vanishing of reduced 1-cohomology for Banach representations

A theorem of Delorme states that every unitary representation of a connected Lie group with nontrivial reduced first cohomology has a finite-dimensional subrepresentation. More recently Shalom showed that such a property is inherited by cocompact lattices and stable under coarse equivalence among amenable countable discrete groups. We give a new geometric proof of Delorme's theorem which extends to a larger class of groups, including solvable $p$-adic algebraic groups, and finitely generated solvable groups with finite Prüfer rank. Moreover all our results apply to isometric representations in a large class of Banach spaces, including reflexive Banach spaces. As applications, we obtain an ergodic theorem in for integrable cocycles, as well as a new proof of Bourgain's Theorem that the 3-regular tree does not embed quasi-isometrically into any superreflexive Banach space.

math.GR↗
arXiv2017

Densely related groups

We study the class of densely related groups. These are finitely generated (or more generally, compactly generated locally compact) groups satisfying a strong negation of being finitely presented, in the sense that new relations appear at all scales. Here, new relations means relations that do not follow from relations of smaller size. Being densely related is a quasi-isometry invariant among finitely generated groups. We check that a densely related group has none of its asymptotic cones simply connected. In particular a lacunary hyperbolic group cannot be densely related. We prove that the Grigorchuk group is densely related. We also show that a finitely generated group that is (infinite locally finite)-by-cyclic and which satisfies a law must be densely related. Given a class $\mathcal{C}$ of finitely generated groups, we consider the following dichotomy: every group in $\mathcal{C}$ is either finitely presented or densely related. We show that this holds within the class of nilpotent-by-cyclic groups and the class of metabelian groups. In contrast, this dichotomy is no longer true for the class of $3$-step solvable groups.

math.GR↗