** "Geometry and Analysis on Groups" Research Seminar **

**Title:**
Hyperbolic rigidity of higher rank lattices

**Speaker:**
Thomas Haettel (Montpellier)

**Abstract:**
Higher rank lattices, like \(SL(n,Z)
for \(n\)
at least 3, enjoy numerous rigidity properties: every isometric action on a tree or on a Hilbert space has a fixed point, for instance. We will show that every action by isometries of a higher rank lattice on a Gromov-hyperbolic space is elementary. Among consequences, we obtain another proof of the Farb-Kaimanovich-Masur Theorem that any morphism from a higher rank lattice to a mapping class group has finite image. Guirardel and Horbez also deduce another proof of the Bridson-Wade theorem that any morphism from a higher rank lattice to
\(\mathrm{Out}(F_n)\)
has finite image.