Abstract: A dendrite is a compact metrizable space such that any two points are connected by a unique arc. Dendrites may appear as Julia sets, Berkovich projective lines and played in important role in the proof of the cut point conjecture for boundaries of hyperbolic groups by Bowditch.

In a common work with Nicolas Monod, we study groups acting on dendrites by homeomorphisms. In this purely topological context, we obtain rigidity results for lattices of algebraic groups and an analog of Tits alternative.

In this talk, I will try to emphasize the large elementary part of the theory.