Alireza Salehi Golsefidy

Expansion in perfect groups

Let Γ be a subgroup of

Local spectral gap in simple Lie groups and applications

{{\rm GL}_d(\mathbb{Z}[1/q_0])}

Counting lattices in simple Lie groups: The positive characteristic case

generated by a finite symmetric set S. For an integer q, denote by πq the projection map

Super-approximation, II: the p-adic and bounded power of square-free integers cases

{\mathbb{Z}[1/q_0] \to \mathbb{Z}[1/q_0]/q \mathbb{Z}[1/q_0]}

On capacity achieving property of rotational coding for acyclic deterministic wireless networks

. We prove that the Cayley graphs of πq(Γ) with respect to the generating sets πq(S) form a family of expanders when q ranges over square-free integers with large prime divisors if and only if the connected component of the Zariski-closure of Γ is perfect, i.e. it has no nontrivial Abelian quotients.

Translate of horospheres and counting problems

We describe all the discrete subgroups of Ad(G0)(F ) o Aut(F ) which act transitively on the set of vertices of B = B(F;G0) the Bruhat-Tits building of a pair (F;G0) of a characteristic zero non-archimedean local eld and a simply-connected absolutely almost simple F -group if B is of dimension at least 4. In fact, we classify all the maximal such subgroups. We show that there are exactly eleven families of such subgroups and explicitly construct them. Moreover, we show that four of these families act simply transitively on the vertices. In particular, we show that there is no such actions if either the dimension of the building is larger than 7, F is not isomorphic to Qp for some prime p, or the building is associated to SLn;D0 where D0 is a non-commutative division algebra. Along the way we also give a new proof of Siegel-Klingen theorem on the rationality of certain Dedekind zeta functions and L-functions. 1