  Divergence functions of a metric space estimate the length of a path
connecting two points $A$, $B$ at distance $\le n$ avoiding a large enough ball
around a third point $C$. We characterize groups with non-linear divergence
functions as groups having cut-points in their asymptotic cones. By
Olshanskii-Osin-Sapir, that property is weaker than the property of having
Morse (rank 1) quasi-geodesics. Using our characterization of Morse
quasi-geodesics, we give a new proof of the theorem of Farb-Kaimanovich-Masur
that states that mapping class groups cannot contain copies of irreducible
lattices in semi-simple Lie groups of higher ranks. It also gives a
generalization of the result of Birman-Lubotzky-McCarthy about solvable
subgroups of mapping class groups not covered by the Tits alternative of Ivanov
and McCarthy.
  We show that any group acting acylindrically on a simplicial tree or a
locally compact hyperbolic graph always has "many" periodic Morse
quasi-geodesics (i.e. Morse elements), so its divergence functions are never
linear. We also show that the same result holds in many cases when the
hyperbolic graph satisfies Bowditch's properties that are weaker than local
compactness. This gives a new proof of Behrstock's result that every
pseudo-Anosov element in a mapping class group is Morse.
  On the other hand, we conjecture that lattices in semi-simple Lie groups of
higher rank always have linear divergence. We prove it in the case when the
$\mathbb{Q}$-rank is 1 and when the lattice is $SL_n(\mathcal{O}_S)$ where
$n\ge 3$, $S$ is a finite set of valuations of a number field $K$ including all
infinite valuations, and $\mathcal{O}_S$ is the corresponding ring of
$S$-integers.
