Spectral properties of finite groups

Thesis

This thesis concerns the diameter and spectral gap of finite groups. Our focus shall be on the asymptotic behaviour of these quantities for sequences of finite groups arising as quotients of a fixed infinite group.

In Chapter 3 we give new upper bounds for the diameters of finite groups which do not depend on a choice of generating set. Our method exploits the commutator structure of certain profinite groups, in a fashion analogous to the Solovay-Kitaev procedure from quantum computation. We obtain polylogarithmic upper bounds for the diameters of finite quotients of: groups with an analytic structure over a pro-p domain (with exponent depending on the dimension); Chevalley groups over a pro-p domain (with exponent independent of the dimension) and the Nottingham group of a finite field. We also discuss some consequences of our results for random walks on groups.

In Chapter 4 we construct new examples of expander Cayley graphs of finite groups, arising as congruence quotients of non-elementary subgroups of SL₂(𝔽p[t]) modulo certain square-free ideals. We describe some applications of our results to simple random walks on such subgroups, specifically giving bounds on the rate of escape from algebraic subvarieties, the set of squares and the set of elements with reducible characteristic polynomial in SL₂(𝔽p[t])

Finally, in Chapter 5 we produce new expander congruence quotients of SL₂ (ℤ_p), generalising work of Bourgain and Gamburd. The proof combines the Solovay-Kitaev procedure with a quantitative analysis of the algebraic geometry of these groups, which in turn relies on previously known examples of expanders.

Authors: Bradford, H

• DOI: 10.5287/ora-n1rr8xqk4
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford