2021/10/06 by Jitendra Bajpai, Bajpai, Jitendra, Daniele Dona +3
Mathematics · Computer Science · #Limits and Structures in Graph Theory #Advanced Mathematical Modeling in Engineering #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2110.02942
Babai's conjecture states that, for any finite simple non-abelian group G, the diameter of G is bounded by (log|G|)C for some absolute constant C. We prove that, for any untwisted classical group G of rank r defined over a field \mathbbFq with q not too small with respect to r, diam(G(\mathbbFq))≤(log|G(\mathbbFq)|)^408r4. This bound improves on results by Breuillard, Green, and Tao [9], Pyber and Szabó [38], and, for q large enough, also by Halasi, Maróti, Pyber, and Qiao [16]. Our approach is in several ways closer to that of preexistent work by Helfgott [20], in that we give dimensional estimates (that is, bounds of the form |A∩ V(\mathbbFq)|≪|AC|dim(V)/dim(G), where A is any generating set) for varieties V of specific types, and work in the Lie algebra whenever possible. One of our main tools is a new, more efficient form of escape from subvarieties.