Acta Mathematica

Volume 221 (2018)

Number 1

Character bounds for finite groups of Lie type

Pages: 1 – 57



Roman Bezrukavnikov (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Ma., U.S.A.)

Martin W. Liebeck (Department of Mathematics, Imperial College, London, United Kingdom)

Aner Shalev (Institute of Mathematics, Hebrew University, Jerusalem, Israel)

Pham Huu Tiep (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.)


We establish new bounds on character values and character ratios for finite groups $G$ of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form $\lvert \chi(g) \rvert \leqslant c \chi (1)^{\alpha g}$, and give rise to a variety of applications, for example to covering numbers and mixing times of random walks on such groups. In particular, we deduce that, if $G$ is a classical group in dimension $n$, then, under some conditions on $G$ and $g \in G$, the mixing time of the random walk on $G$ with the conjugacy class of $g$ as a generating set is (up to a small multiplicative constant) $n/s$, where $s$ is the support of $g$.

The first author was partially supported by the NSF grants DMS-1102434 and DMS-1601953. The second and third authors acknowledge the support of EPSRC grant EP/H018891/1. The third author acknowledges the support of ERC advanced grant 247034, ISF grants 1117/13 and 686/17, BSF grant 2016072 and the Vinik chair of mathematics which he holds. The fourth author was partially supported by the NSF grants DMS-1839351 and DMS-1840702, the Simons Foundation Fellowship 305247, the EPSRC, and the Mathematisches Forschungsinstitut Oberwolfach.

Received 2 November 2017

Received revised 7 August 2018

Accepted 10 August 2018

Published 6 November 2018