Mathieu Carette

Publications and preprints

The Haagerup property is not invariant under quasi-isometry. With an appendix by Sylvain Arnt, Thibault Pillon, Alain Valette
Preprint (6 pages), 2014. pdf | arXiv | abstract
Abstract: Using the work of Cornulier-Valette and Whyte, we show that neither the Haagerup property nor weak amenability is invariant under quasi-isometry of finitely generated groups.
Commability of groups quasi-isometric to trees.
Preprint (22 pages), 2013. pdf | arXiv | abstract
Abstract:
Commability is the finest equivalence relation between locally compact groups such that G and H are equivalent whenever there is a continuous proper homomorphism G -> H with cocompact image. Answering a question of Cornulier, we show that all non-elementary locally compact groups acting geometrically on locally finite simplicial trees are commable, thereby strengthening previous forms of quasi-isometric rigidity for trees. We further show that 6 homomorphisms always suffice, and provide the first example of a pair of locally compact groups which are commable but without commation consisting of less than 6 homomorphisms. Our strong quasi-isometric rigidity also applies to products of symmetric spaces and Euclidean buildings, possibly with some factors being trees.
Virtually splitting the map from Aut(G) to Out(G).
To appear in Proc. Amer. Math. Soc. (12 pages), 2013. pdf | arXiv | abstract
Abstract:
We give an elementary criterion on a group G for the map from Aut(G) to Out(G) to split virtually. This criterion applies to many residually finite CAT(0) groups and hyperbolic groups, and in particular to all finitely generated Coxeter groups. As a consequence the outer automorphism group of any finitely generated Coxeter group is residually finite and virtually torsion-free.
Locally compact convergence groups and n-transitive actions. (with Dennis Dreesen)
Preprint (31 pages), 2012. pdf | arXiv | abstract
Abstract:
All sigma-compact, locally compact groups acting sharply n-transitively and continuously on compact spaces M have been classified, except for n=2,3 when M is infinite and disconnected. We show that no such actions exist for n=2 and that these actions for n=3 coincide with the action of a hyperbolic group on a space equivariantly homeomorphic to its hyperbolic boundary. We further give a characterization of non-compact groups acting 3-properly and transitively on infinite compact sets as non-elementary boundary transitive hyperbolic groups. The main tool is a generalization to locally compact groups of Bowditch's topological characterization of hyperbolic groups. Finally, in contrast to the case n=3, we show that for n>3, if a locally compact group acts continuously, n-properly and n-cocompactly on a locally connected metrizable compactum M, then M has a local cut point.
Spectral rigidity of automorphic orbits in free groups. (with Stefano Francaviglia, Ilya Kapovich and Armando Martino)
Algebraic and Geometric Topology, vol. 12 (2012), pp. 1457-1486. pdf | arXiv | abstract
Abstract:
It is well-known that a point T in the (unprojectivized) Culler-Vogtmann Outer space cv_N is uniquely determined by its translation length function ||.||_T : F_N → R. A subset S of a free group F_N is called spectrally rigid if, whenever T,T' in cv_N are such that ||g||_T=||g||_T' for every g in S then T=T' in cv_N. By contrast to the similar questions for the Teichmüller space, it is known that for N > 1 there does not exist a finite spectrally rigid subset of F_N.
In this paper we prove that for N > 2 if H < Aut(F_N) is a subgroup that projects to an infinite normal subgroup in Out(F_N) then the H-orbit of an arbitrary nontrivial element g in F_N is spectrally rigid. We also establish a similar statement for F₂=F(a,b), provided that g in F₂ is not conjugate to a power of [a,b].

Corrigendum to "Spectral rigidity of automorphic orbits in free groups". (with Stefano Francaviglia, Ilya Kapovich and Armando Martino)
To appear in Algebraic and Geometric Topology (6 pages), 2013. pdf | arXiv | abstract
Abstract:
In this corrigendum we give a corrected proof of Lemma 5.1 about the existence of a fully irreducible element in an infinite normal subgroup of Out(F_N). Our original proof of Lemma 5.1 relied on a subgroup classification result of Handel-Mosher, originally stated by Handel-Mosher for arbitrary subgroups H<Out(F_N). After our paper was published, it turned out that the proof of the Handel-Mosher subgroup classification theorem needs the assumption that H be finitely generated. The corrigendum provides an alternative proof of Lemma 5.1 which uses the corrected, finitely generated, version of the Handel-Mosher theorem and relies on the 0-acylindricity of the action of Out(F_N) on the free factor complex (due to Bestvina-Mann-Reynolds). A proof of 0-acylindricity is included in the corrigendum.
On the rank of Coxeter groups. (with Richard Weidmann)
Preprint (36 pages), 2009. pdf | arXiv | abstract
Abstract:
We show that the standard generating set of a Coxeter group is of minimal cardinality provided that the non-diagonal entries of the Coxeter matrix are sufficiently large.
The automorphism group of accessible groups.
J. Lond. Math. Soc. (2) 84 (2011), no.3, 731-748. pdf | arXiv | abstract
Abstract:
In this article, we study the outer automorphism group of a group G decomposed as a finite graph of group with finite edge groups and finitely generated vertex groups with at most one end. We show that Out(G) is essentially obtained by taking extensions of relative automorphism groups of vertex groups, groups of Dehn twists and groups of automorphisms of free products. We apply this description and obtain a criterion for Out(G) to be finitely presented, as well as a necessary and sufficient condition for Out(G) to be finite. Consequences for hyperbolic groups are discussed.

Formation

2006 - 2009:	Doctorat à l'Université libre de Bruxelles, Belgique Directeurs : Bernhard Mühlherr et Richard Weidmann Thèse : Le groupe d'automorphismes des groupes accessibles et le rang des groupes de Coxeter Jury : Frédéric Bourgeois (Président), Pierre-Emmanuel Caprace, Dimitri Leemans (Secrétaire), Gilbert Levitt.
2002 - 2006:	Licence en sciences mathématiques, Université libre de Bruxelles, Belgique. La Plus Grande Distinction

Postes occupés

2010 - 2014:	Chargé de Recherches du Fonds de la Recherche Scientifique - FNRS à l'Université catholique de Louvain, Belgique
Nov 07 - Jan 08:	Boursier pré-doctoral Marie Curie à l'Université de Münster, Allemagne
2006 - 2010:	Aspirant du Fonds de la Recherche Scientifique - FNRS à l'Université libre de Bruxelles, Belgique

Séjours

Sept 13 - Sept 14:	McGill University, Montréal, Canada
Nov 12:	Centre de Recerca Matematica, Belaterra, Espagne
Août - Sept 10:	Centre de Recherches Mathématiques, Montréal, Canada
Oct 09 - Juin 10:	University of Illinois at Urbana-Champaign, IL, USA
Plusieurs mois entre 06 - 09:	Heriot-Watt University, Edimbourg, UK

Exposés de conférences

Déc 13	Canadian Mathematical Society Winter Meeting - Session on Geometric Group Theory and Low Dimensional Topology, Ottawa, Canada.
Mai 13	Random Matrices, Quantum Hall Effect, Geometric Group Theory, Corfou, Grèce. Minicours : Hyperbolic groups and Kazhdan's property (T)
Nov 12	Automorphisms of free groups - Cool-down week, Belaterra, Espagne.
Oct 12	Buildings 2012, Münster, Allemagne.
Jan 12	Young Geometric Group Theory Meeting, Bedlewo, Pologne.
Déc 11	KIT Weihnachtsworkshop, Karlsruhe, Allemagne.
Oct 11	Buildings 2011, Münster, Allemagne.
Mai 11	Discrete groups and geometric structures IV, Oostende, Belgique.
Oct 09	Buildings 2009, Münster, Allemagne.
Juin 09	Geometric group theory : Davis 60, Bedlewo, Pologne.
Mai 08	Discrete groups and geometric structures III, Kortrijk, Belgique.
Oct 06	Buildings 2006, Münster, Allemagne.

Exposés de séminaires

Mars 14	Séminaire d'Analyse Harmonique, Orsay, France.
Jan 14	Séminaire Géométrie Dynamique, Lille; Séminaire Théorie Ergodique, Rennes; Séminaire Théorie spectrale et géométrie, Grenoble; Séminaire Géométries, Lyon, France.
Oct 13	McGill group theory seminar, Montréal, Canada.
Juin 13	Groupe de travail de théorie géométrique des groupes, Orsay, France.
Juin 13	Operator Algebra Seminar, KULeuven, Belgique.
Mars 13	Séminaire de Théorie Ergodique, Rennes, France.
Déc 12	Séminaire KUL-UCL : Ergodic actions of higher rank lattices d'après D. Creutz et J. Peterson, Leuven, Belgique.
Juin 12	Group theory seminar, UCLouvain, Belgique.
Nov 11	Groupe de travail KUL-UCL : Amenability of topological full groups d'après R. Grigorchuk et K. Medynets, KULeuven, Belgique.
Déc 10	Groupe de travail : Mostow rigidity, UCLouvain, Belgique.
Déc 10	Séminaire Géométrie, Algèbre et Algèbres d'opérateurs, Clermont-Ferrand, France.
Juil 10	Geometry seminar, CAUniversität zu Kiel, Allemagne.
Avr 10	UIUC Group Theory Seminar, Urbana, USA.
Mars 09	ULB-UGent-VUB Incidence Geometry Seminar, Bruxelles, Belgique.
Nov 08	Heriot-Watt algebra seminar, Edinburgh, UK.
May 08	ULB-UGent-VUB Incidence Geometry Seminar, Bruxelles, Belgique.
Jan 08	Oberseminar Geometrie, Universität Münster, Allemagne.
Nov 06	Heriot-Watt algebra seminar, Edinburgh, UK.

2012-2013:	Introduction à la théorie géométrique des groupes (M1-M2)
2011-2012:	TD de mathématiques générales (L1)
2010-2011:	Introduction à la théorie géométrique des groupes (M1-M2) TD de mathématiques générales (L2)

2008-2009:	TD de mathématiques générales (L1)
2007-2008:	TD de géométrie projective (L3) TD de théorie des nombres (L2)
2006-2007:	TD de géométrie projective (L3) TD de théorie des nombres (L2)