Let G be a higher rank semisimple linear algebraic group over a non-Archimedean local field. The simplicial complexes corresponding to any sequence of pairwise non-conjugate irreducible lattices in G are Benjamini-Schramm convergent to the Bruhat-Tits building. Convergence of the relative Plancherel measures and normalized Betti numbers follows. This extends the work [ABB + 17] from real Lie groups to linear groups over arbitrary local fields. Along the way, various results concerning Invariant Random Subgroups and in particular a variant of the classical Borel density theorem are also extended.

Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about v v such manifolds of volume at most v, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, almost all hyperbolic manifolds are non-arithmetic in an appropriate sense. Moreover, by restricting attention to non-compact manifolds, our result implies the same growth type for the number of quasiisometry classes of lattices in SO(n, 1). Our method involves a geometric graph-of-spaces construction that relies on arithmetic properties of certain quadratic forms.

We establish topological local rigidity for uniform lattices in compactly generated groups, extending the result of Weil from the realm of Lie groups. We generalize the classical local rigidity theorem of Selberg, Calabi and Weil to irreducible uniform lattices in Isom(X) where X is a proper CAT(0) space with no Euclidian factors, not isometric to the hyperbolic plane. We deduce an analog of Wang's finiteness theorem for certain non-positively curved metric spaces.

We show that surface groups are flexibly stable in permutations. Our method is purely geometric and relies on an analysis of branched covers of hyperbolic surfaces. Along the way we establish a quantitative variant of the LERF property for surface groups which may be of independent interest.

Let X be a proper geodesic Gromov hyperbolic metric space and let G be a cocompact group of isometries of X admitting a uniform lattice. Let d be the Hausdorff dimension of the Gromov boundary ∂X. We define the critical exponent δ(µ) of any discrete invariant random subgroup µ of the locally compact group G and show that δ(µ) > d 2 in general and that δ(µ) = d if µ is of divergence type. Whenever G is a rank-one simple Lie group with Kazhdan's property (T ) it follows that an ergodic invariant random subgroup of divergence type is a lattice. One of our main tools is a maximal ergodic theorem for actions of hyperbolic groups due to Bowen and Nevo.