Higher Rank Semisimple Groups Research Articles

Approximate Lattices in Higher-Rank Semi-Simple Groups

Approximate Lattices in Higher-Rank Semi-Simple Groups

On Benjamini-Schramm limits of congruence subgroups

A sequence of orbifolds corresponding to pairwise non-conjugate congruence lattices in a higher rank semisimple group over zero characteristic local fields is Benjamini-Schramm convergent to the universal cover.

Stabilizers of actions of lattices in products of groups

We prove that any ergodic non-atomic probability-preserving action of an irreducible lattice in a semisimple group, with at least one factor being connected and of higher-rank, is essentially free. This generalizes the result of Stuck and Zimmer [Stabilizers for ergodic actions of higher rank semisimple groups.Ann. of Math. (2)139(3) (1994), 723–747], who found that the same statement holds when the ambient group is a semisimple real Lie group and every simple factor is of higher-rank. We also prove a generalization of a result of Bader and Shalom [Factor and normal subgroup theorems for lattices in products of groups.Invent. Math.163(2) (2006), 415–454] by showing that any probability-preserving action of a product of simple groups, with at least one having property$(T)$, which is ergodic for each simple subgroup, is either essentially free or essentially transitive. Our method involves the study of relatively contractive maps and the Howe–Moore property, rather than relying on algebraic properties of semisimple groups and Poisson boundaries, and introduces a generalization of the ergodic decomposition to invariant random subgroups, which is of independent interest.

Arithmetic Groups, Base Change, and Representation Growth

Consider an arithmetic group $${\mathbf{G}(O_S)}$$ , where $${\mathbf{G}}$$ is an affine group scheme with connected, simply connected absolutely almost simple generic fiber, defined over the ring of S-integers O S of a number field K with respect to a finite set of places S. For each $${n \in \mathbb{N}}$$ , let $${R_n(\mathbf{G}(O_S))}$$ denote the number of irreducible complex representations of $${\mathbf{G}(O_S)}$$ of dimension at most n. The degree of representation growth $${\alpha(\mathbf{G}(O_S)) = \lim_{n \rightarrow\infty} \log R_n(\mathbf{G}(O_S)) / \log n}$$ is finite if and only if $${\mathbf{G}(O_S)}$$ has the weak Congruence Subgroup Property. We establish that for every $${\mathbf{G}(O_S)}$$ with the weak Congruence Subgroup Property the invariant $${\alpha(\mathbf{G}(O_S))}$$ is already determined by the absolute root system of $${\mathbf{G}}$$ . To show this we demonstrate that the abscissae of convergence of the representation zeta functions of such groups are invariant under base extensions $${K{\subset}L}$$ . We deduce from our result a variant of a conjecture of Larsen and Lubotzky regarding the representation growth of irreducible lattices in higher rank semi-simple groups. In particular, this reduces Larsen and Lubotzky’s conjecture to Serre’s conjecture on the weak Congruence Subgroup Property, which it refines.

A group theoretical characterisation of S-arithmetic groups in higher rank semi-simple groups

We give a characterisations for an abstract group to be an S-arithmetic group of a higher rank semi-simple group.

Quasi-isometric classification of non-uniform lattices in semisimple groups of higher rank

We give another proof of the quasi-isometric classification theorem of non-uniform lattices in higher rank semisimple groups. We use the asymptotic cone and a class of ats (the logarithmic ats) which move away in the cusp with logarithmic speed.

Finitely generated profinitely dense free groups in higher rank semi-simple groups

We prove that if Γ is an arithmetic subgroup of a non-compact linear semi-simple groupG such that the associated simply connected algebraic group over ℚ has the so-called congruence subgroup property, then Γ contains a finitely generated profinitely dense free subgroup. As a corollary we obtain af·g·p·d·f subgroup of SL n (ℤ) (n ≧ 3. More generally, we prove that if Γ is an irreducible arithmetic non-cocompact lattice in a higher rank group, then Γ containsf·g·p·d·f groups.