Distortion in Cremona groups

Abstract

We study distortion of elements in 2-dimensional Cremona groups over algebraically closed fields of characteristic zero. Namely, we obtain the following trichotomy: non-elliptic elements (i.e., those whose powers have unbounded degree) are undistorted; elliptic elements have a doubly exponential, or exponential distortion according to whether they are virtually unipotent.By definition, δ c,S (n) = ∞ if and only if c has finite order. Clearly, if S ⊂ T then δ c,S ≤ δ c,T . Also, δ c,S k (n) = δ c,S (kn). In particular, if S ⊂ T k , then δ c,S ≤ δ c,T (kn). If S T , it follows that δ c,S δ c,T , and if S ≃ T then δ c,S ≃ δ c,T .If S and T both generate G then S ≃ T and δ c,S ≃ δ c,T . Thus, when G is finitely generated, the ≃-equivalence class of the distorsion function only depends on (G, c), not on the finite generating subset; it is is called the distortion function of c in G, and is denoted δ G c , or simply δ c . The element c is called undistorted if δ c (n) n, and distorted otherwise.Example 1.1. Fix a pair of integers k, ℓ ≥ 2. In the Baumslag-Solitar groupIn the "double" Baumslag-Solitar group, one finds double exponential distorsion (see [22] and § 8).It is natural to consider distortion in groups which are not finitely generated. We say that an element c ∈ G is undistorted if δ H c (n) ≃ n for every finitely generated subgroup H of G containing c. Changing H may change the distorsion function δ H c ; for instance, if c is not a torsion element, it is undistorted in H = c Z but may be distorted in larger groups. Also, there are examples of pairs (G, c) such that c becomes more and more distorted, in larger and larger subgroups of G (see § 8). Thus, we have a good notion of distorsion, but the distorsion is not measured by an equivalence class of a function " δ G c ". We shall say that the distorsion type (or class) of c in G is at least f if there is finitely generated subgroup H containing c with f δ H c , and at most g if δ H c g for all finitely generated subgroup H containing c. If the distorsion type is at least f and at most f simultaneously, we shall say that f is the distorsion type of c. For instance, c may be exponentially, or doubly exponentially distorted in G.Example 1.2. Let k be a field. Let c be an element of the general linear group GL d (K); we have one of the following (see [25,24] and § 3)