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Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective

Published online by Cambridge University Press:  15 April 2019

C. M. Campbell
Affiliation:
University of St Andrews, Scotland
C. W. Parker
Affiliation:
University of Birmingham
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
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Summary

By now, we have a product theorem in every finite simple group G of Lie type, with the strength of the bound depending only in the rank of G. Such theorems have numerous consequences: bounds on the diameters of Cayley graphs, spectral gaps, and so forth. For the alternating group Altn, we have a quasipolylogarithmic diameter bound (Helfgott-Seress 2014), but it does not rest on a product theorem. We shall revisit the proof of the bound for Altn, bringing it closer to the proof for linear algebraic groups, and making some common themes clearer. As a result, we will show how to prove a product theorem for Altn – not of full strength, as that would be impossible, but strong enough to imply the diameter bound.

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Publisher: Cambridge University Press
Print publication year: 2019

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