Algebraic actions of discrete groups: the $p$-adic method

Abstract

The first argument is based on p-adic analysis and may be viewed as an extension of two classical strategies from a linear to a non-linear context. The first strategy appeared in the proof of the theorem of Skolem, Mahler, and Lech, which says that the zeros of a linear recurrence sequence occur along a finite union of arithmetic progressions. This method plays now a central role in arithmetic dynamics (see [6] , [7], [55]). The second strategy has been developed by Bass, Milnor, and Serre when they obtained rigidity results for finite-dimensional linear representations of SL n (Z) as a corollary of the congruence subgroup property (see [2], [62]). Here, we combine these strategies for non-linear actions of finitely generated groups of birational transformations.Our second argument makes use of isoperimetric inequalities from geometric group theory and of the Lang-Weil estimates from diophantine geometry. We now list the main results that follow from the combination of those arguments. Actions of SL n (Z)Consider the group SL n (Z) of n×n matrices with integer entries and determinant 1. Let Γ be a finite-index subgroup of SL n (Z); it acts by linear projective transformations on the projective space P n−1 C , and the kernel of the homomorphism Γ!PGL n (C) contains at most two elements. The following result shows that Γ does not act faithfully on any smaller variety.

Regularization of birational actions of FW groups

“…The assertions of the corollary (for X reduced in characteristic zero) are known for groups with Kazhdan's Property T [8], using analytic methods. Nevertheless, Theorem 1.1 is new also in the case when G has Kazhdan's Propery T, say with X smooth over C.…”

Entropy, Lyapunov exponents, and rigidity of group actions

“…Early results establishing this conjecture in the setting of actions the circle appear in [42,93,208] and in the setting of volume-preserving (and more general measure-preserving) actions on surfaces in [87,88,165]. See also [92] and [74] for results on real-analytic actions and [44,46,47] for results on holomorphic and birational actions. There are also many results (usually in the C 0 setting) for actions of specific lattices on manifolds where there are topological obstructions to the group acting; a partial list of such results includes [29,30,162,206,207,211,212,221].…”

Actions of nilpotent groups on complex algebraic varieties

“…Proof. Theorem 2.11 of [CX18] shows that ι extends uniquely to a continuous map. In [CX18] this is only shown when p ≥ 3 and c = 1 but the proof is identical with p ≥ 2 and c > 1…”

“…Proposition 4.2 (Proposition 4.4 of [BGT10], Proposition 3.2 of [CX18]). Let X be an irreducible complex quasi-projective variety, α ∈ X(C) and Γ be a finitely generated subgroup of AutC(X).…”