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 Bowditchs properties that are weaker than local compactness. This gives a new proof of Behrstocks 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 $nge 3$, $S$ 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.
Following Gromov, the coboundary expansion of building-like complexes is studied. In particular, it is shown that for any $n geq 1$, there exists a constant $epsilon(n)>0$ such that for any $0 leq k <n$ the $k$-th coboundary expansion constant of any $n$-dimensional spherical building is at least $epsilon(n)$.
We study cocompact lattices with dense projections in a product $G_1 times G_2$ of locally compact groups and show, under the assumption that each $G_i$ is a closed subgroup of the automorphism group $Aut(T_i)$ of a regular tree satisfying certain local transitivity conditions, that such a lattice is contained in only finitely many discrete subgroups of $G_1 times G_2$.
We confirm a conjecture of Jens Marklof regarding the equidistribution of certain sparse collections of points on expanding horospheres. These collections are obtained by intersecting the expanded horosphere with a certain manifold of complementary dimension and turns out to be of arithmetic nature. This equidistribution result is then used along the lines suggested by Marklof to give an analogue of a result of W. Schmidt regarding the distribution of shapes of lattices orthogonal to integer vectors.
