 
Summary: UNDISTORTED SOLVABLE LINEAR GROUPS
HERBERT ABELS AND ROGER ALPERIN
Abstract. We discuss distortion of solvable linear groups over a lo
cally compact 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] gives a natural condition for a subgroup to be undistorted using its eigen
values: no eigenvalues of absolute value one in the adjoint representation.
Gromov claims that this sufficient condition is also necessary; however, it is
not and in fact one only needs this condition for the representation on the
abelianization of the unipotent subgroup. Our main theorem (3.2) then con
cerns distortion in linear solvable groups over a characteristic zero local field
and makes precise the necessary and sufficient conditions that a subgroup
is undistorted.
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. nonnegativity, symmetry, triangle inequality,
zero on the diagonal, but not necessarily the property that d(x, y) = 0
