TY - JOUR

T1 - Character bounds for regular semisimple elements and asymptotic results on Thompson’s conjecture

AU - Larsen, Michael

AU - Taylor, Jay

AU - Tiep, Pham Huu

N1 - Funding Information:
The first author was partially supported by the NSF grant DMS-2001349. The third author gratefully acknowledges the support of the NSF (Grants DMS-1840702 and DMS-2200850), the Joshua Barlaz Chair in Mathematics, and the Charles Simonyi Endowment at the Institute for Advanced Study (Princeton). The authors are grateful to the referee for careful reading and helpful comments on the paper, in particular for the suggestion to add Corollary 2 to the paper.
Publisher Copyright:
© 2023, The Author(s).

PY - 2023/2

Y1 - 2023/2

N2 - For every integer k there exists a bound B= B(k) such that if the characteristic polynomial of g∈ SL n(q) is the product of ≤ k pairwise distinct monic irreducible polynomials over Fq, then every element x of SL n(q) of support at least B is the product of two conjugates of g. We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions (p, q), in the special case that n= p is prime, if g has order qp-1q-1, then every non-scalar element x∈ SL p(q) is the product of two conjugates of g. The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.

AB - For every integer k there exists a bound B= B(k) such that if the characteristic polynomial of g∈ SL n(q) is the product of ≤ k pairwise distinct monic irreducible polynomials over Fq, then every element x of SL n(q) of support at least B is the product of two conjugates of g. We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions (p, q), in the special case that n= p is prime, if g has order qp-1q-1, then every non-scalar element x∈ SL p(q) is the product of two conjugates of g. The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.

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U2 - 10.1007/s00209-022-03193-3

DO - 10.1007/s00209-022-03193-3

M3 - Article

AN - SCOPUS:85146773738

SN - 0025-5874

VL - 303

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 2

M1 - 47

ER -