A numerical invariant for finitely generated groups via actions on graphs.
A remark on the intersection of the conjugates of the base of quasi-HNN groups.
A very short proof of Forester's rigidity result.
A weak type (1,1) estimate for a maximal operator on a group of isometries of a homogeneous tree
We give a simple proof of a result of R. Rochberg and M. H. Taibleson that various maximal operators on a homogeneous tree, including the Hardy-Littlewood and spherical maximal operators, are of weak type (1,1). This result extends to corresponding maximal operators on a transitive group of isometries of the tree, and in particular for (nonabelian finitely generated) free groups.
Actions de groupes sur les arbres
Actions of finitely generated groups on -trees
We study actions of finitely generated groups on -trees under some stability hypotheses. We prove that either the group splits over some controlled subgroup (fixing an arc in particular), or the action can be obtained by gluing together actions of simple types: actions on simplicial trees, actions on lines, and actions coming from measured foliations on -orbifolds. This extends results by Sela and Rips-Sela. However, their results are misstated, and we give a counterexample to their statements.The...
Amenability of linear-activity automaton groups
We prove that every linear-activity automaton group is amenable. The proof is based on showing that a random walk on a specially constructed degree 1 automaton group – the mother group – has asymptotic entropy 0. Our result answers an open question by Nekrashevych in the Kourovka notebook, and gives a partial answer to a question of Sidki.
Amenable actions of nonamenable groups.
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 semiregular trees acting doubly...
Applications harmoniques entre graphes finis et un théorème de superrigidité
Applications harmoniques entre graphes finis et un théorème de superrigidité
Nous définissons une ntoion d’énergie pour des applications entre deux graphes métriques finis et cherchons à minimiser l’énergie au sein d’une classe d’homotopie. Nous démontrons des théorèmes d’existence et d’unicité analogues à ceux de Eells-Sampson et de Hartman pour les applications harmoniques à valeurs dans les variétés à courbure négative ou nulle. Nous montrons également une propriété de stabilité des applications minimisantes par rapport aux revêtements de degré fini à la source. Une application...
Automorphism groups of right-angled buildings: simplicity and local splittings
We show that the group of type-preserving automorphisms of any irreducible semiregular thick right-angled building is abstractly simple. When the building is locally finite, this gives a large family of compactly generated abstractly simple locally compact groups. Specialising to appropriate cases, we obtain examples of such simple groups that are locally indecomposable, but have locally normal subgroups decomposing non-trivially as direct products, all of whose factors are locally normal.
Baumslag-Solitar groups and some other groups of cohomological dimension two.
Bounding the complexity of simplicial group actions on trees.
Cœur et nombre d'intersection pour les actions de groupes sur les arbres
Commensurability of graph products.
Conformally non-sperical 2-complexes.
Conservative polynomials and yet another action of on plane trees
In this paper we study an action of the absolute Galois group on bicolored plane trees. In distinction with the similar action provided by the Grothendieck theory of “Dessins d’enfants” the action is induced by the action of on equivalence classes of conservative polynomials which are the simplest examples of postcritically finite rational functions. We establish some basic properties of the action and compare it with the Grothendieck action.
Construction du cœur compact d’un arbre réel par substitution d’arbre
Étant donné un automorphisme d’un groupe libre et un représentant topologique train-track de son inverse, on peut construire un arbre réel appelé arbre répulsif de . Le groupe libre agit sur par isométries. La dynamique engendrée par peut être représentée par l’action du groupe libre restreinte à un sous-ensemble compact bien choisi du complété métrique de . Cet article construit ce sous-ensemble sur une classe d’exemples en introduisant des opérations appelées substitutions d’arbre ;...
Contractibility of deformation spaces of -trees.