Abstract

Bounded languages are a class of formal languages which includes all context free languages of polynomial growth. We prove that if a finitely generated group G admits a combing by a bounded language and this combing satisfies the asynchronous fellow traveller property, then either G is virtually abelian, or else G contains an element g of infinite order such that gn and gm are conjugate for some 0 < n < m.

The introduction of automatic groups [5] has precipitated a host of questions about the roles which formal language theory and geometry play in the study of normal forms for finitely generated groups, particularly groups which arise in geometric settings. For example, when a group G is given as the fundamental group of a compact Riemannian manifold, words in a fixed set of generators for G have a natural interpretation as paths in the universal cover of the manifold; it is natural to ask how the geometry of the manifold is reflected in the linguistic complexity of normal forms for elements of G. The results presented here and in [3] can be interpreted as providing a partial answer to this question in the case where the manifold under consideration is a quotient of a nilpotent Lie group.