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 |χ(g)|⩽cχ(1)αg, 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∈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.
We associate to any given finite set of valuations on the polynomial ring in two variables over an algebraically closed field a numerical invariant whose positivity characterizes the case when the intersection of their valuation rings has maximal transcendence degree over the base fields. As an application, we give a criterion for when an analytic branch at infinity in the affine plane that is defined over a number field in a suitable sense is the branch of an algebraic curve.
Qing LuSchool of Mathematical Sciences, Beijing Normal UniversityWeizhe ZhengMorningside Center of Mathematics and Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
Arithmetic Geometry and Commutative Algebramathscidoc:1904.07002