TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 363, Number 6, June 2011, Pages 3185–3210 S 0002-9947(2011)05237-2 Article electronically published on January 3, 2011
UNDISTORTED SOLVABLE LINEAR GROUPS
HERBERT ABELS AND ROGER ALPERIN
Abstract. We discuss distortion of solvable linear groups over a locally com- pact field and provide necessary and sufficient conditions for a subgroup to be undistorted when the field is of characteristic zero.
1. Introduction Distortion of subgroups of solvable groups is the main object of our study. We concentrate on the case of linear solvable groups. In this case Gromov [6, 3. D2 Remark] gives a natural condition for a subgroup to be undistorted using its eigen- values: no eigenvalues of absolute value one in the adjoint representation (see Ex- ample 6.6 below for a discussion of this condition). We give necessary and sufficient conditions in our main theorem, Theorem 3.2, that a linear solvable group over a characteristic zero local field is undistorted in its ambient general linear group.
2. Preliminaries 2.1. A pseudometric on X is a function d : X ×X → R which has all the properties of a metric, i.e., non–negativity, symmetry, triangle inequality, and zero on the diagonal, but not necessarily the property that d(x, y) = 0 implies x = y.A pseudometric space is a pair (X, d) consisting of a set X and a pseudometric d on X.
2.2. Let (X, dX )and(Y,dY ) be pseudometric spaces. A map f : X → Y is called a submetry if there are real constants C1 > 0andC2 such that dY (f(x),f(x )) ≤ C1dX (x, x )+C2 for every pair x, x of points of X.IfX = Y andtheidentitymapofX is a submetry from (X, d1)to(X, d2), we write
d1 ≺ d2
and say that d1 is dominated by d2. A submetry f :(X, dX ) → (Y,dY ) is called a quasi–isometry if there is a submetry g :(Y,dY ) → (X, dX ) such that both g◦f and f ◦ g are of bounded distance from the identity, i.e., if there is a real constant C3 such that dX (x, g ◦ f(x)) ≤ C3 and dY (y, f ◦ g(y)) ≤ C3 for every x ∈ X and every y ∈ Y . This definition is equivalent to the usual definition of a quasi–isometry f
Received by the editors July 29, 2009. 2000 Mathematics Subject Classification. Primary 20G99, 20G25, 22E99, 22E25. The authors thank the SFB701 at Bielefeld and also MSRI for partial support during the period of research and preparation of this paper.
c 2011 by the authors 3185
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from (X, dX )to(Y,dY ); namely requiring that there be real constants C1 > 0and C2 such that −1 − ≤ ≤ C1 dX (x, x ) C2 dY (f(x),f(x )) C1dX (x, x )+C3 for every pair x, x of points in X and that there be, for every y ∈ Y , a point x ∈ X such that dY (y, f(x)) ≤ C2. Since then the map g mapping y ∈ Y to such an x ∈ X gives the required quasi–inverse g from (Y,dY )to(X, dx).
Two pseudometrics d1 and d2 on the same set X are called quasi–equivalent if the identity map of X is a quasi–isometry from (X, d1)to(X, d2), or from (X, d2) to (X, d1), which is the same condition. Thus d1 and d2 are quasi–equivalent iff they dominate each other, i.e., d1 ≺ d2 and d2 ≺ d1.Wethenwrite d1 ≺ d2. 2.3. The word metric. Let G be a locally compact topological group. Suppose G has a compact set E of generators. Let E be the corresponding word length,so { ε1 εq ∈ ∈{ − }} E (g)=min q ; g = a1 ...aq with ai E and εi +1, 1 . −1 Let dE be the corresponding left invariant metric; that is, dE (g, h)= E(g h). The metric dE is called the word metric for the set E of generators of G.Itiswell known that the word metrics dE and dE are quasi–equivalent if E and E are sets of generators of G which are either compact or relatively compact neighborhoods of the identity. We thus have a unique quasi–equivalence class of pseudometrics on G, and we call every pseudometric in this quasi–equivalence class a (or sometimes even the) word metric on G. Whenever we speak of G as a pseudometric space, we mean one of these pseudometrics, unless a special warning is given. 2.4. The word metric d is a proper metric. Thismeansthatforeveryx ∈ G and every R ∈ R the ball BR(x)={g ∈ G ; d(g, x)