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 (Zp), 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:712392 |
| Date | January 2015 |
| Creators | Bradford, Henry |
| Contributors | Lackenby, Mark |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://ora.ox.ac.uk/objects/uuid:10babab2-8d11-4d53-aea8-7479b868a57d |
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