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Andrés Navas

Publications and source records attributed to Andrés Navas.

At least 37 records · Page 2Linked to original sources
arXiv2019

Uniformly perfect finitely generated simple left orderable groups

We show that the finitely generated simple left orderable groups $G_ρ$ constructed by the first two authors in arXiv:1807.06478 are uniformly perfect - each element in the group can be expressed as a product of three commutators of elements in the group. This implies that the group does not admit any homogeneous quasimorphism. Moreover, any nontrivial action of the group on the circle, which lifts to an action on the real line, admits a fixed point. Most strikingly, it follows that the groups are examples of left orderable monsters, which means that any faithful action on the real line without a global fixed point is globally contracting. This answers Question 4 from the 2018 ICM proceedings article of the third author. (This question has also been answered simultaneously and independently, using completely different methods, by Matte Bon and Triestino in arXiv:1811.12256.) To prove our results, we provide a certain characterisation of elements of the group $G_ρ$ which is a useful new tool in the study of these examples.

math.GR↗
arXiv2018

Groups with infinitely many ends acting analytically on the circle

This article takes the inspiration from two milestones in the study of non minimal actions of groups on the circle: Duminy's theorem about the number of ends of semi-exceptional leaves and Ghys' freeness result in analytic regularity. Our first result concerns groups of analytic diffeomorphisms with infinitely many ends: if the action is non expanding, then the group is virtually free. The second result is a Duminy's theorem for minimal codimension one foliations: either non expandable leaves have infinitely many ends, or the holonomy pseudogroup preserves a projective structure.

math.DS↗
arXiv2018

Group actions on 1-manifolds: a list of very concrete open questions

This text focuses on actions on 1-manifolds. We present a (non exhaustive) list of very concrete open questions in the field, each of which is discussed in some detail and complemented with a large list of references, so that a clear panorama on the subject arises from the lecture.

math.DS↗
arXiv2016

On the ergodic theory of free group actions by real-analytic circle diffeomorphisms

We consider finitely generated groups of real-analytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are only finitely many orbits of connected components of its complement. For the case of minimal actions, we show that if the underlying group is (algebraically) free, then the action is ergodic with respect to the Lebesgue measure. This provides first answers to questions due to É. Ghys, G. Hector and D. Sullivan.

math.DS↗
arXiv2015

The amazing story of a forgotten golden flag

We describe the most probable geometric design of the Chilean Independence Flag, which uses the golden ratio in many of its components. We also discuss some related historical aspects.

math.HO↗
arXiv2015

Non-rectifiable Delone sets in SOL and other solvable groups

Given a lattice $Γ\subset SOL$, we show that there is a coarsely dense subset $\mathcal{D} \subset Γ$ that is not biLipschitz equivalent to $Γ$. We also prove similar results for lattices in certain higher rank abelian-by-abelian groups and for the solvable Baumslag-Solitar groups.

math.MG↗
arXiv2015

Some examples of non-rectifiable, repetitive Delone sets

We construct examples of Delone sets of the plane (that is, discrete subsets that are uniformly separated and coarsely dense) that are repetitive (each patch of the set appears in every large-enough ball) though non-rectifiable (i.e. non bi-Lipschitz equivalent to the standard lattice). More generally, we construct such a set so that the translation action on the closure of its orbit has any prescribed Choquet simplex as its set of invariant probability measures (in particular, we provide uniquely ergodic examples). The construction relies on classical examples of Burago-Kleiner and McMullen, for which we give pure combinatorial (an effective) versions that have interest by themselves.

math.MG↗
arXiv2014

Groups, orders, and laws

We discuss the question whether left-orderable groups satisfying a nontrivial law are locally indicable.

math.GR↗
arXiv2013

Sur les rapprochements par conjugaison en dimension 1 et classe C^1

We prove that the space of actions of Z^d by C^1 (orientation-preserving) diffeomorphisms of either the interval or the circle is connected by arcs. This is proved by showing that all such actions can be C^0 conjugated via a 1-parameter family into diffeomorphisms that converge to either the trivial action or an action by Euclidean rotations. Extensions for nilpotent group actions are provided.

math.DS↗
arXiv2013

Almost reduction and perturbation of matrix cocycles

In this note, we show that if all Lyapunov exponents of a matrix cocycle vanish, then it can be perturbed to become cohomologous to a cocycle taking values in the orthogonal group. This extends a result of Avila, Bochi and Damanik to general base dynamics and arbitrary dimension. We actually prove a fibered version of this result, and apply it to study the existence of dominated splittings into conformal subbundles of general matrix cocycles.

math.DS↗
arXiv2013

A geometric path from zero Lyapunov exponents to rotation cocycles

We consider cocycles of isometries on spaces of nonpositive curvature $H$. We show that the supremum of the drift over all invariant ergodic probability measures equals the infimum of the displacements of continuous sections under the cocycle dynamics. In particular, if a cocycle has uniform sublinear drift, then there are almost invariant sections, that is, sections that move arbitrarily little under the cocycle dynamics. If, in addition, $H$ is a symmetric space, then we show that almost invariant sections can be made invariant by perturbing the cocycle.

math.DS↗
arXiv2011

An L^1 ergodic theorem with values in a nonpositively curved space via a canonical barycenter map

We extend a recent result of Tim Austin (see arXiv:0905.0515) to the L^1 setting, thus providing a general version of the Birkhoff ergodic theorem for functions taking values in nonpositively curved spaces. In this setting, the notion of a Birkhoff sum is replaced by that of a barycenter along the orbit. The construction of an appropriate barycenter map is the core of this note. In particular, we solve a problem raised by K.-T. Sturm showing that local compactness for the underlying space is superfluous for the construction (this extends a result of A. Es-Sahib and H. Heinich). As a byproduct of our construction, we prove a fixed point theorem for actions by isometries on a Buseman space.

math.DS↗
arXiv2011

A Livsic type theorem for germs of analytic diffeomorphisms

We deal with the problem of the validity of Livsic's theorem for cocycles of diffeomorphisms satisfying the orbit periodic obstruction over an hyperbolic dynamics. We give a result in the positive direction for cocycles of germs of analytic diffeomorphisms at the origin.

math.DS↗
arXiv2011

Reduction of cocycles and groups of diffeomorphisms of the circle

We prove two theorems of reduction of cocycles taking values in the group of diffeomorphisms of the circle. They generalise previous results obtained by the author concerning rigidity for smooth actions on the circle of Kazhdan's groups and higher rank lattices.

math.RT↗