Group actions on dendrites and curves

Duchesne, Bruno; Monod, Nicolas

Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 2277-2309.

Nous établissons des obstructions à l’existence d’actions de groupes par homéomorphismes sur des dendrites. Par exemple, les réseaux de groupes de Lie simples en rang supérieur à 2 fixent toujours un point ou une paire de points. Le même résultat est obtenu pour des réseaux irréductibles dans des produits de groupes connexes. D’autres résultats incluent une alternative de Tits et une description de la dynamique des actions de groupes.

Nous discutons brièvement dans quelle mesure nos résultats peuvent s’étendre au cas plus général des courbes topologiques.

We establish obstructions for groups to act by homeomorphisms on dendrites. For instance, lattices in higher rank simple Lie groups will always fix a point or a pair. The same holds for irreducible lattices in products of connected groups. Further results include a Tits alternative and a description of the topological dynamics.

We briefly discuss to what extent our results hold for more general topological curves.

Classification : 54H20, 22D40
Keywords: dendrites, groups, rigidity, bounded cohomology, lattices, Tits alternative, dynamics, curves
Mot clés : dendrites, groupes, rigidité, cohomology bornée, réseaux, alternative de Tits, dynamique, courbes

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