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Growth and expansion in algebraic groups over finite fields

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... We do not answer this specific question, but rather study what happens when a is an arbitrary element of F p (as opposed to being the mod p quotient of a fixed value a ∈ Z independent of p as in the above mentioned works). It was observed by Bukh, Harper, Helfgott and Lindenstrauss (see [10,Exercises 3.13 and 3.14]), that a general polylogarithmic upper bound on the mixing time, valid for all primes p and for all a of sufficiently large multiplicative order, can be deduced easily from some estimates of Konyagin [15]. This gives (1) ) (1.2) provided the multiplicative order of a is at least O(log p log log p) in F p . ...
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We study the Markov chain xn+1=axn+bnx_{n+1}=ax_n+b_n on a finite field Fp{\mathbb {F}}_p, where aFp×a \in {\mathbb {F}}_p^{\times } is fixed and bnb_n are independent and identically distributed random variables in Fp{\mathbb {F}}_p. Conditionally on the Riemann hypothesis for all Dedekind zeta functions, we show that the chain exhibits a cut-off phenomenon for most primes p and most values of aFp×a \in {\mathbb {F}}_p^\times . We also obtain weaker, but unconditional, upper bounds for the mixing time.
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For every connected, almost simple linear algebraic group GGLnG\le \textrm{GL}_{n} over a large enough field K, every subvariety VGV\subseteq G, and every finite generating set AG(K)A\subseteq G(K), we prove a general dimensional bound, that is, a bound of the form with C1C_{1}, C2C_{2} depending only on n, deg(V)\textrm{deg}(V). The dependence of C1C_1 on n (or rather on dim(V)\dim (V)) is doubly exponential, whereas C2C_2 (which is independent of deg(V)\textrm{deg}(V)) depends simply exponentially on n. Bounds of this form have proved useful in the study of growth in linear algebraic groups since 2005 (Helfgott) and, before then, in the study of subgroup structure (Larsen–Pink: A a subgroup). In bounds for general V and G available before our work, the dependence of C1C_1 and C2C_2 on n was of exponential-tower type. We draw immediate consequences regarding diameter bounds for untwisted classical groups G(Fq)G(\mathbb {F}_{q}). (In a separate paper, we derive stronger diameter bounds from stronger dimensional bounds we prove for specific families of varieties V.)
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