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New bounds on growth in SL_n

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Harald Helfgott (University of Göttingen)
Rudradip Biswas (University of Manchester/MPIM)
Thu, 2021-07-29 16:00 - 17:00
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(joint work with Jitendra Bajpai and Daniele Dona)

Abstract: Let G be a finite group. For A\subset G a set of generators, the diameter diam \Gamma(G,A) is the minimal r such that (A \cup A^{-1}\cup \{e})^r = G. Define the diameter of G, diam(G), to be the maximum of diam \Gamma(G,A) over all sets of generators A.

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. Following work of mine proving the conjecture for G=SL_2(F_p) and SL_3(F_p), Breuillard-Green-Tao and Pyber-Szabo proved that, for G a finite simple group of Lie type, the diameter of G is bounded by (\log |G|)^{C_n}, where C_n depends on n. The dependence was not made explicit, but their proofs surely give C_n growing faster than exponentially on n (namely, exp(n^{O(1)}).

We give a proof for G = SL_n(F_q) giving diam(G)\leq (\log |G|)^{C_n} with C_n polynomial, namely, C_n = O(n^4 \log n). This bound should be compared not just with Breuillard-Green-Tao and Pyber-Szabo but also with the bound diam(G)\leq q^{O(n (\log n)^2)} (Halasi-Maróti-Pyber-Qiao, 2019). Our proof is based on an improved escape procedure and on dimensional estimates particular to conjugacy classes and abelian subgroups.

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