This is an informal introductory talk about geometrical invariants I like the most. I'll give a survey on the minimal entropy conjecture on symmetric spaces and its connections. If time permits, I'll present also a possible relation to systolic geometry.
We will discuss estimating word distance between two geometric outer automorphisms of the free group \Out(F_n) using the geometry of the hyperbolic simplicial complexes it acts on. We will present the construction of the obtained distance formula for \Out(F_n) through its analogy with the similar formulas for the special linear group and the mapping class group. This is a joint work with Yulan Qing and Kasra Rafi.