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

PY - 2023/1/23

Y1 - 2023/1/23

N2 - For every integer k there exists a bound B=B(k) such that if the characteristic polynomial of g∈SLn(q) is the product of ≤k pairwise distinct monic irreducible polynomials over Fq, then every element x of SLn(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∈SLp(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∈SLn(q) is the product of ≤k pairwise distinct monic irreducible polynomials over Fq, then every element x of SLn(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∈SLp(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.

UR - http://www.scopus.com/inward/record.url?scp=85146773738&partnerID=8YFLogxK

U2 - 10.1007/s00209-022-03193-3

DO - 10.1007/s00209-022-03193-3

M3 - Article

SN - 0025-5874

VL - 303

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 2

M1 - 47

ER -