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Elementary embeddings in torsion-free hyperbolic groups

Chloé Perin (2011)

Annales scientifiques de l'École Normale Supérieure

We describe first-order logic elementary embeddings in a torsion-free hyperbolic group in terms of Sela’s hyperbolic towers. Thus, if H embeds elementarily in a torsion free hyperbolic group Γ , we show that the group Γ can be obtained by successive amalgamations of groups of surfaces with boundary to a free product of  H with some free group and groups of closed surfaces. This gives as a corollary that an elementary subgroup of a finitely generated free group is a free factor. We also consider the...

Finitude géométrique en géométrie de Hilbert

Mickaël Crampon, Ludovic marquis (2014)

Annales de l’institut Fourier

On étudie la notion de finitude géométrique pour certaines géométries de Hilbert définies par un ouvert strictement convexe à bord de classe  𝒞 1 .La définition dans le cadre des espaces Gromov-hyperboliques fait intervenir l’action du groupe discret considéré sur le bord de l’espace. On montre, en construisant explicitement un contre-exemple, que cette définition doit être renforcée pour obtenir des définitions équivalentes en termes de la géométrie de l’orbifold quotient, similaires à celles obtenues...

Fixed points of endomorphisms of certain free products

Pedro V. Silva (2012)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

The fixed point submonoid of an endomorphism of a free product of a free monoid and cyclic groups is proved to be rational using automata-theoretic techniques. Maslakova’s result on the computability of the fixed point subgroup of a free group automorphism is generalized to endomorphisms of free products of a free monoid and a free group which are automorphisms of the maximal subgroup.

Fixed points of endomorphisms of certain free products

Pedro V. Silva (2012)

RAIRO - Theoretical Informatics and Applications

The fixed point submonoid of an endomorphism of a free product of a free monoid and cyclic groups is proved to be rational using automata-theoretic techniques. Maslakova’s result on the computability of the fixed point subgroup of a free group automorphism is generalized to endomorphisms of free products of a free monoid and a free group which are automorphisms of the maximal subgroup.

Flats in Spaces with Convex Geodesic Bicombings

Dominic Descombes, Urs Lang (2016)

Analysis and Geometry in Metric Spaces

In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales.We show that some such results remain valid for metric spaces with non-unique geodesic segments under suitable convexity assumptions on the distance function along distinguished geodesics. The discussion includes, among other things, the Flat Torus Theorem and Gromov’s hyperbolicity criterion referring to embedded planes. This generalizes...

Gradients de Heegaard sous-logarithmiques d’une variété hyperbolique de dimension trois et fibres virtuelles

Claire Renard (2010/2011)

Séminaire de théorie spectrale et géométrie

J. Maher a montré qu’une variété hyperbolique de dimension 3 compacte sans bord, connexe et orientable fibre virtuellement sur le cercle si et seulement si elle admet une famille infinie de revêtements finis de genre de Heegaard borné. En s’appuyant sur la démonstration de Maher, cet article présente un théorème donnant une condition suffisante pour qu’un revêtement fini d’une variété hyperbolique compacte de dimension 3 contienne une fibre virtuelle, qui s’exprime en fonction du degré d du revêtement...

Groupes aléatoires

Étienne Ghys (2002/2003)

Séminaire Bourbaki

Quelles sont les propriétés d’un groupe de présentation finie “tiré au hasard” ? La réponse à cette question dépend bien entendu de la méthode choisie pour le tirage au sort. On peut par exemple fixer n générateurs et choisir p relations aléatoirement parmi les mots de longueur L , puis faire tendre L vers l’infini. On peut aussi choisir un graphe fini, étiqueter aléatoirement ses arêtes par des générateurs, et considérer le groupe engendré par ces générateurs, soumis aux relations lues sur les cycles...

Groups of real analytic diffeomorphisms of the circle with a finite image under the rotation number function

Yoshifumi Matsuda (2009)

Annales de l’institut Fourier

We consider groups of orientation-preserving real analytic diffeomorphisms of the circle which have a finite image under the rotation number function. We show that if such a group is nondiscrete with respect to the C 1 -topology then it has a finite orbit. As a corollary, we show that if such a group has no finite orbit then each of its subgroups contains either a cyclic subgroup of finite index or a nonabelian free subgroup.

Harmonic measures versus quasiconformal measures for hyperbolic groups

Sébastien Blachère, Peter Haïssinsky, Pierre Mathieu (2011)

Annales scientifiques de l'École Normale Supérieure

We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as quasiconformal measures on the boundary of the group.

Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions

Elena Fuchs, Chen Meiri, Peter Sarnak (2014)

Journal of the European Mathematical Society

We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature ( n - 1 , 1 ) is “thin”, namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg’s theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many hyperbolic hypergeometric groups for n F n - 1 are thin.

Isometries of systolic spaces

Tomasz Elsner (2009)

Fundamenta Mathematicae

We provide a classification of isometries of systolic complexes corresponding to the classification of isometries of CAT(0)-spaces. We prove that any isometry of a systolic complex either fixes the barycentre of some simplex (elliptic case) or stabilizes a thick geodesic (hyperbolic case). This leads to an alternative proof of the fact that finitely generated abelian subgroups of systolic groups are undistorted.

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