INTRODUCTION

Let ℓ be a length function on a group G such that N, the set of non-Archimedean elements, is a proper subset of G. Then Theorem 5.3 of shows that the lengths of elements of N are bounded if and only if ℓ(ax) = ℓ(x) for all a in N, x in G\N, in which case N is a subgroup of G and ℓ is an extension of a non-Archimedean length function on N by an Archimedean length function on G/N. Among other results, this allows the structure of any length function on an abelian group to be determined.

This leaves the question of whether, if N is a proper subgroup of G, the lengths of elements of N are necessarily bounded. In this paper we show that the answer is no. Theorem 1 of §1 gives a relation between ℓ(ax), ℓ(a) and ℓ(x), for a in N, x in G\ N, whenever N is a proper subgroup of G whose elements have unbounded lengths. This suggests that it may be possible to express such a length function in terms of a non-Archimedean length function ℓ1 = ℓ∣N on N and an Archimedean length function ℓ2 on G/N. In §2 we give an example of an HNN-group G with a length function ℓ, such that N is a proper subgroup with unbounded lengths.