Logarithmic diameter for Cayley graphs of some finitely generated matrix group

Abstract

In the 1980s Nori and Weisfeiler independently discovered the strong approximation property of a wide range of groups. Afterwards, a stronger version called super approximation has developed and has been a fast growing topic in the recent decades. The strong approximation may be realised as that for finitely generated matrix group Γ = ⟨S⟩ ⊂ GLn(Z), its Zariski closure in (GLn)Z and its closure in the profinite group GLn(̂Z ) are, roughly speaking, two equivalent measures of the size of Γ. The super approximation is concerned with an even stronger property. The group Γ is said to have super approximation property with respect to C if the group not only has strong approximation, but the family of Cayley graphs Xq ∶= Cay(πq(Γ), πq(S)), q ∈ C(C being some subset of N, πq being the modulo q reduction) is an expander. The latter condition of an expander family {Xq}q∈C means that Xq’s are sparse graphs with uniformly strong connectivity, and in particular have the uniform logarithmic diameter bound. The geometric structure underlying the Cayley graphs of a group is closely related to its group expansion properties. Thus, the super approximation property of a group assures the logarithmic diameter bound for the associated Cayley graphs Xq. Among the various recent breakthroughs toward super approximation, one important contribution due to Salehi Golsefidy, is the super approximation conjecture, which asserts that the group Γ has super approximation with respect to N if and only if the connected component of its Zariski closure is perfect. In this thesis, I provide a quite strong supporting evidence to this unsolved super approximation conjecture. Specifically, the logarithmic diameter bound for family of Cayley graphs {Xq}q∈N of a finitely generated matrix group Γ ⊂ GLn(Z) is established when the Zariski closure Zcl(Γ) of Γ is connected and perfect, which can also be rephrased as that Zcl(Γ) is a semidirect product of a semisimple group and its unipotent radical. To its end, I will give expressions of the important results and tools for deriving logarithmic diameter, including the bounded generation for semisimple group and its action on vector group (which is the abelianization of unipotent radical), bounded generation of perfect group, and the p-adic quantitative open image theorems developed by Salehi Golsefidy. Then I will discuss the new case, i.e. not yet in the literature, of a perfect Zariski closure where the unipotent radical is not necessarily abelian. I will start with the suggestive special case that the unipotent radical is Heisenberg group, and then by extending the idea, settle the general situation.